Patent Publication Number: US-6987282-B2

Title: Quantum bit with a multi-terminal junction and loop with a phase shift

Description:
RELATED APPLICATIONS 
     This application is related to concurrently filed application Ser. No. 09/839,637 entitled “Quantum Bit with a Multi-Terminal Junction and Loop with a Phase Shift” and application Ser. No. 09/839,991 entitled “Quantum Bit with a Multi-Terminal Junction and Loop with a Phase Shift”, both of which are herein incorporated by reference in their entirety. 
    
    
     BACKGROUND 
     1. Field of the Invention 
     This invention relates to quantum computing and, more specifically, to solid state quantum computing qubits with superconducting materials. 
     2. Discussion of Related Art 
     Research on what is now called quantum computing traces back to Richard Feynman, See, e.g., R. Feynman,  Int. J. Theor. Phys ., 21, 467-488 (1982). Feynman noted that quantum systems are inherently difficult to simulate with conventional computers but that observing the evolution of a quantum system could provide a much faster way to solve some computational problems. In particular, solving a theory for the behavior of a quantum system commonly involves solving a differential equation related to the Hamiltonian of the quantum system. Observing the behavior of the quantum system provides information regarding the solutions to the equation. 
     Further efforts in quantum computing were initially concentrated on “software development” or building of the formal theory of quantum computing. Software development for quantum computing involves attempting to set the Hamiltonian of a quantum system to correspond to a problem requiring solution. Milestones in these efforts were the discoveries of the Shor and Grover algorithms. See, e.g., P. Shor,  SIAM J. of Comput ., 26:5, 1484-1509 (1997); L. Grover, Proc. 28th STOC, 212-219 (1996); and A. Kitaev, LANL preprint quant-ph/9511026 (1995). In particular, the Shor algorithm permits a quantum computer to factorize natural numbers. Showing that fault-tolerant quantum computation is theoretically possible opened the way for attempts at practical realizations of quantum computers. See, e.g., E. Knill, R. Laflamme, and W. Zurek,  Science , 279, p. 342 (1998). 
     One proposed application of a quantum computer is the factoring of large numbers. In such an application, a quantum computer could render obsolete all existing encryption schemes that use the “public key” method. In another application, quantum computers (or even a smaller scale device such as a quantum repeater) could enable absolutely safe communication channels where a message, in principle, cannot be intercepted without being destroyed in the process. See, e.g., H. J. Briegel et al., LANL preprint quant-ph/9803056 (1998) and the references therein. 
     Quantum computing generally involves initializing the states of N qubits (quantum bits), creating controlled entanglements among the N qubits, allowing the quantum states of the qubit quantum system to evolve under the influence of the entanglements, and reading the qubits after they have evolved. A qubit quantum system is conventionally a system having two degenerate quantum states, where the state of the qubit quantum system can have non-zero probability of being found in either degenerate state. Thus, N qubit quantum systems can define an initial state that is a combination of 2 N  states. The entanglements between qubits and the interactions between the qubits and external influences control the evolution of the distinguishable quantum states and define calculations that the evolution of the quantum states perform. This evolution, in effect, can perform 2 N  simultaneous calculations. Reading the qubits after evolution is complete determines the states of the qubit quantum systems and the results of the calculations. 
     Several physical systems have been proposed for the qubits in a quantum computer. One system uses chemicals having degenerate nuclear spin states, see U.S. Pat. No. 5,917,322, “Method and Apparatus for Quantum Information Processing”, to N. Gershenfeld and I. Chuang. Nuclear magnetic resonance (NMR) techniques can read the spin states. These systems have successfully implemented a search algorithm, see, e.g., J. A. Jones, M. Mosca, and R. H. Hansen “Implementation of a Quantum Search Algorithm on a Quantum Computer,”  Nature , 393, 344-346 (1998) and the references therein, and a number ordering algorithm, see, e.g., Lieven M. K. Vandersypen, Matthias Steffen, Gregory Breyta, Costantino S. Yannoni, Richard Cleve and Isaac L. Chuang, “Experimental Realization of Order-Finding with a Quantum Computer,” LANL preprint quant-ph/0007017 (2000),  Phys. Rev. Lett . , Vol. 85, No. 25, 5452-55 (2000) and the references therein. The number ordering algorithm is related to the quantum Fourier transform, an essential element of both Shor&#39;s algorithm for factoring of a natural number and Grover&#39;s Search Algorithm for searching unsorted databases, see T. F. Havel, S. S. Somaroo, C.-H. Tseng, and D. G. Cory, “Principles and Demonstrations of Quantum Information Processing by NMR Spectroscopy, 2000, ” LANL preprint quant-ph/9812086 V2 (1999), and the references therein. However, efforts to expand such systems to a commercially useful number of qubits face difficult challenges. 
     Another physical system for implementing a qubit includes a superconducting reservoir, a superconducting island, and a dirty Josephson junction that can transmit a Cooper pair (of electrons) from the reservoir into the island. The island has two degenerate states. One state is electrically neutral, but the other state has an extra Cooper pair on the island. A problem with this system is that the charge of the island in the state having the extra Cooper pair causes long range electric interactions that interfere with the coherence of the state of the qubit. The electric interactions can force the island into a state that definitely has or lacks an extra Cooper pair. Accordingly, the electric interactions can end the evolution of the state before calculations are complete or qubits are read. This phenomenon is commonly referred to as collapsing the wavefunction, loss of coherence, or decoherence. See Y. Nakamura, Yu. A. Pashkin and J. S. Tsai “Coherent Control of Macroscopic Quantum States in a Single-Cooper-Pair Box,”  Nature  V. 398 No. 6730, P.786-788 (1999), and the references therein. 
     Another physical system for implementing a qubit includes a radio frequency superconducting quantum interference device (RF-SQUID). See J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, and Seth Lloyd, “Josephson Persistent-Current Qubit,”  Science  285, 1036-39 (Aug. 13, 1999), and the references therein. The energy levels of the RF-SQUID correspond to differing amounts of magnetic flux threading the SQUID ring. Application of a static magnetic field normal to the SQUID ring may bring two of these energy levels, corresponding to different magnetic fluxes threading the ring, into resonance. Typically, external AC magnetic fields are also applied to pump the system into excited states so as to maximize the tunneling frequency between qubit basis states. A problem with this system is that the basis states used are not naturally degenerate and the required biasing field has to be extremely precise. This biasing is possible for one qubit, but with several qubits, this bias field fine-tuning becomes extremely difficult. Another problem is that the basis states used are typically not the ground states of the system, but higher energy states populated by external pumping. This requires the addition of an AC field generating device, whose frequency will differ for each qubit as the individual qubit parameters vary. 
     The race to create the first scalable, practical, and powerful solid state quantum computer has existed for over ten years. Ever since the notion of a quantum computer first became evident with Feynman in 1982, scientists have been creating qubits of various forms. There are currently a number of disclosed qubits, where the quantum states are realized in the doubly degenerate ground states of the flux in a superconducting loop. Inevitably, these qubits are only useful when controlled by magnetic fields, or by some other means which couple the qubit to the environment or provide other potential sources of decoherence. In order to overcome these sources of decoherence, a large amount of overhead is required to control and harvest the quantum power available from the qubit. However, the means by which this can be accomplished has as yet eluded scientists. Thus, there is a need for a qubit which does not require the coupling magnetic fields, but which can be controlled by applying and reading currents and voltages. 
     There therefore exists a need for integrated solid state structures that can form the basic building blocks out of which integrated circuits using quantum effects can be built. The desired structures are such that they can be read from, written to and operated on in an efficient and scalable manner. 
     SUMMARY 
     In accordance with the present invention, a qubit is comprised of a multi-terminal junction, where two of the terminals of the junction are directly connected together, thus forming a superconducting loop. The superconducting loop introduces a phase shift so that the phase of the superconducting order parameter Ψ is shifted by απ in transition through the structure, where α ranges from −1, through zero (no phase shift), to 1. A phase shift can be produced, for example, by the inclusion of a phase shifter in the superconducting loop or by external application of a magnetic field through the superconducting loop. 
     A qubit according to the present invention can be constructed from a multi-terminal Josephson junction in which at least two terminals of the junction are coupled to a superconducting loop to form a superconducting loop and at least two further terminals are open and can be coupled to external current sources. The multi-terminal junction can be made of superconducting leads coupled, for example, by any combination of constriction junctions (also referred to as micro-bridges), tunnel junctions, or semiconducting two dimensional electron gas structures within the physical location. In some embodiments of the qubit, the terminals of the multi-terminal junction are coupled in a physical location whose size is less than the size of the qubit. 
     In some embodiments of the invention, properties of both a symmetric junction and an asymmetric junction can be utilized. In a symmetric junction, a change in the direction of the transport current in the junction equally affects current in the terminals of the superconducting loop, thus having no overall affect on the current in the loop. In an asymmetric junction, a change in the direction of the transport current differentially affects the terminals that form the superconducting loop, thus changing the overall current in the loop. 
     A symmetric junction allows for the reduction of the potential energy barrier between the two nearly degenerate ground states of the quantum system of the qubit, thus providing a means of applying a σ x  quantum gate operation. An asymmetric junction allows for biasing of one of the two ground states of the qubit, thus providing a means of applying a σ z  quantum gate operation. 
     A phase shifter is any structure that shifts the phase of the superconducting order parameter Ψ by απ in transition through the structure, where α is a constant such that −1≦α≦1. The phase shift in the superconducting loop causes time-reversal symmetry breakdown in the qubit quantum system and thus causes a double degeneracy of the ground state without requiring an external magnetic flux or other influence. In some embodiments, the terminals in a multi-terminal junction can be physically asymmetric. This asymmetry affects the properties of a qubit according to the present invention by controlling the phase shift of the order parameter Ψ in transition through a multi-terminal junction. 
     A qubit according to the present invention may be constructed out of any superconducting material. Embodiments of qubits having any desired number of terminals and a phase shifter can also be constructed in accordance with desired applications for the qubit. Embodiments of qubit structures include, for example, s-wave superconductor/normal metal/d-wave superconductor/normal metal/s-wave superconductor, referred to as S-N-D-N-S junctions, superconductor/ferromagnet/superconductor, referred to as S-F-S junctions, s-wave superconductor/two dimensional electron gas/s-wave superconductor, referred to as S-2DEG-S junctions, or multi-crystal d-wave superconductors patterned on an insulating substrate. The equilibrium ground state of the qubit quantum system is, in the absence of external magnetic fields, twice degenerate, with one of the energy levels corresponding to a magnetic flux threading the loop in one sense (corresponding to an equilibrium supercurrent flow, for example, in the clockwise direction around the superconducting loop), and the other energy level corresponding to a magnetic flux threading the loop in the opposite sense (corresponding to an equilibrium supercurrent flow, for example, in the counterclockwise direction around the superconducting loop). 
     Some embodiments of qubits according to the present invention include an s-wave (for example, niobium, aluminum, lead, mercury, or tin) superconducting structure that includes an asymmetric four-terminal junction with all terminals connected by constriction junctions. Two of the terminals can be joined to form a superconducting loop and the other two terminals can be coupled to a source of transport current. The superconducting loop includes a phase shifter, which may consist of a S-N-D-N-S (for example, niobium/gold/YBa 2 CU 3 O 7-x /gold/nobium) junction. If the incoming current is parallel to the a (or b) crystallographic direction of the d-wave material, and the outgoing current is parallel to the b (or a) crystallographic direction of the d-wave material, this S-N-D-N-S junction can give a phase shift of π. Choosing the incoming and outgoing currents to be at any arbitrary angle to each other in the a-b plane in this embodiment allows a more general phase shift. 
     A magnetic field may also be applied to the superconducting loop. Both the transport current and the external magnetic field may be controlled so as to initialize the state of the qubit quantum system, allow control of the evolution of the qubit quantum system state and read the final state of the qubit quantum system after the evolution (and therefore the desired calculation) is complete. Further, qubits can be selectively entangled by coupling superconducting loops from different qubit structures with a switchable junction, allowing for control of entanglements in a qubit array. 
     A qubit according to the present invention can include a junction with any number of terminals. Some embodiments of the invention include a five terminal junction. A superconducting loop is formed between two terminals of the five terminal junction. The remaining three terminals, two terminals adjacent to the looping terminals, and one terminal centrally opposite the looping terminals, form a means by which to implement all desired quantum operations, including reading, writing, a σ x  gate operation, and a σ z  gate operation. 
     Some embodiments, such as the five-terminal qubit, include both symmetric and asymmetric properties. Because the critical current in the junction depends on the state of the qubit, a read operation can be performed by applying a current asymmetrically across the junction, with a magnitude between the critical currents of the two states and determining if a resistance is created. Additionally, a σ z  gate operation can be performed by applying a pulse of current asymmetrically across the junction while a σ x  gate operation can be performed by applying a pulse of current symmetrically across the junction. 
     In accordance with some embodiments of the invention, a quantum computing method cools a structure containing at least one multi-terminal qubit to a temperature that makes the structure superconducting and suppresses decoherence processes in the system. The actual temperature will depend on the superconducting materials of the qubit. After the structure is at the appropriate temperature, a supercurrent can be established in each superconducting loop, the supercurrent being in a particular classical bit state corresponding to the information input for the quantum calculation. The quantum systems of each of the plurality of qubits is then allowed to evolve in the presence of externally applied magnetic fields and transport currents (whose details constitute the “software” or algorithm being followed by the structure). This allows each superconducting loop bit state to evolve into quantum states that are admixtures of a first state having a first magnetic moment and a second state having a second magnetic moment. These quantum states evolve under the action of the system&#39;s Hamiltonian in the manner prescribed by quantum mechanics. The evolution performs the quantum computation. Determining a measured magnetic moment or flux due to the supercurrent in each superconducting loop determines the result of the quantum computation. 
     In accordance with another aspect of the invention, determining the measured magnetic moments of the quantum state on the qubit can also include applying an alternating transport current and/or magnetic bias field to each qubit and then measuring the magnetic flux produced by the supercurrent flowing in each superconducting loop. In some embodiments, a static transport current and/or magnetic bias field can be applied to each qubit and the voltage across at least two of the terminals in the multi-terminal junction measured to determine the quantum state of the qubit. In some embodiments, the quantum states of the qubits can be read directly with, for example, a SQUID magnetometer. 
     In further aspects of the invention, quantum qubits can be selectively entangled with switchable junctions. In embodiments where qubits include a superconducting loop, an array of qubits can be entangled by switchably coupling the superconducting loops of the array. Additionally, a switchable junction can be included to decouple selected ones of the superconducting loops from other multi-terminal junctions. 
     These and other embodiments according to the present invention are further discussed below with respect to the following figures. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIGS. 1A ,  1 B,  1 D, and  1 F through  1 M show plan views of embodiments of multi-terminal junctions in accordance with embodiments of the present invention. 
         FIGS. 1C and 1E  show cross-sectional views of embodiments of multi-terminal junctions in accordance with embodiments of aspects of the present invention. 
         FIGS. 2A-2G  show example embodiments of phase shifter structures in accordance with aspects of the present invention. 
         FIG. 3A  shows a plan view of a four-terminal qubit with intrinsic phase shifter qubit horizontal architecture in accordance with embodiments of the present invention. 
         FIG. 3B  shows a plan view of a multi-terminal qubit with intrinsic phase shifter qubit horizontal architecture in accordance with embodiments of the present invention. 
         FIG. 3C  shows a plan view of a plurality of multi-terminal junctions connected in a superconducting loop with intrinsic phase shifter qubit horizontal architecture in accordance with embodiments of the present invention. 
         FIG. 4A  shows a plan view of a plurality of multi-terminal qubits, each having a superconducting loop with an intrinsic phase shifter in accordance with embodiments of the present invention. 
         FIG. 4B  shows a plan view of a two-terminal circuit with a plurality of multi-terminal qubits each having a superconducting loop with intrinsic phase shifter in accordance with embodiments of the present invention. 
         FIG. 5  shows a plan view of a six-terminal junction connecting two superconducting loops, each with an intrinsic phase shifter, to form a pair of qubits in accordance with embodiments of the present invention. 
         FIGS. 6A and 6B  show plan views of a plurality of multi-terminal junctions coupling a plurality of superconducting loops with intrinsic phase shifters to form a plurality of qubits in accordance with embodiments of the present invention. 
         FIG. 7  shows a plan view of a voltage measurement circuit in accordance with embodiments of the present invention. 
         FIG. 8  shows a plan view of a five-terminal quantum qubit according to embodiments of the present invention. 
         FIG. 9  shows an exemplary method of entangling a pair of five terminal qubits as shown in FIG.  8 . 
         FIGS. 10   a  and  10   b  illustrate a switchable entanglement method for coupling two five terminal qubits. 
         FIG. 11  shows an array of N five terminal qubits according to the present invention. 
         FIG. 12  shows another embodiment of a qubit array according to the present invention. 
         FIG. 13  shows a plan view of a quantum qubit according to the present invention. 
         FIG. 14.  4-terminal Josephson junctions. (a) Junction with microbridges. (b) Mesoscopic junction with two-dimensional electron gas (2DEG) and symmetric current configuration. In this configuration, coupling between currents I and J inside the normal region is local (δ=0). (c) Mesoscopic junction in the asymmetric currents configuration. Coupling is non-local (δ≠0). 
       FIG.  15 . Mesoscopic 4-terminal SQUID qubit. The π-phase shifter in the flux loop is added to attain bistability without external flux. The externally controlled transport current I affects the current J in the superconducting loop through the phase dragging effect and in turns the flux in the loop. 
       FIG.  16 . Two coupled five-terminal qubits. The left qubit has asymmetric current configuration while the right one has a symmetric one. 
       FIG.  17 . System of N coupled qubits. The distance between the flux regions is maximized to have less magnetic interaction between the qubits. 
       Use of the same reference symbols in different figures indicates elements having similar or identical functions. 
     
    
    
     DETAILED DESCRIPTION 
     In accordance with embodiments of the invention, a quantum computing operation can be performed on an array of quantum qubits where at least one of the qubits includes a qubit according to the present invention. A qubit according to the present invention includes a multi-terminal junction where two terminals of the multi-terminal junction are joined to form a superconducting loop. The superconducting loop introduces a phase shift to the superconducting order parameter. In some embodiments, the superconducting loop includes an intrinsic phase shifter. 
     Intrinsic phase shifters in superconducting phase quantum bits (qubits) are disclosed in M. H. S. Amin, T. Duty, A. Omelyanchouk, G. Rose and A. Zagoskin, U.S. Provisional Application Ser. No. 60/257624, “Intrinsic Phase Shifter as an Element of a Superconducting Phase Quantum Bit”, filed Dec. 22, 2000, herein incorporated by reference in its entirety. A phase shifting structure with 0 and π-phase shifts in a two-terminal DC SQUID is described in R. R. Schulz, B. Chesca, B. Goetz, C. W. Schneider, A. Schmehl, H. Bielefeldt, H. Hilgenkamp, J. Mannhart and C. C. Tsuei, “Design and Realization of an all d-Wave dc π-Superconducting Quantum Interference Device”,  Appl. Phys. Lett.  76, 7 p.912-14 (2000), herein incorporated by reference in its entirety. 
       FIG. 13  shows a block diagram of a qubit  100  according to the present invention. Qubit  100  includes multi-terminal junction  120  with terminals  110 - 1  through  110 -N where N is an integer. At least two of terminals  110 - 1  through  110 -N are coupled to portions  124  and  125  of a superconducting loop  122 . In  FIG. 13 , terminals  110 - 1  and  110 -N are coupled to superconducting loop  122 , but in general any two of terminals  110 - 1  through  110 -N can be coupled to superconducting loop  122 . Superconducting loop  122  further includes a phase shifting structure  123  coupled into superconducting loop  122 . In  FIG. 13 , terminals  110 - i  and  110 - j  refer to arbitrary ones of terminals  110 - 1  through  110 -N. Although terminals  110 - 1  and  110 -N are shown coupled to portions  124  and  125 , respectively, of superconducting loop  122 , in general any pair of terminals  110 - 1  through  110 -N can be coupled to terminals  124  and  125  of superconducting loop  122 . 
     Embodiments of qubits  100  according to the present invention can include phase shifter  123 , which introduces an arbitrary phase shift inclusively between −π and π. Phase shifters, such as phase shifter  123 , that introduce arbitrary phase shifts can be more practical for the construction of qubits since the phase shifters with arbitrary phase shifts are more easily constructed. Further tuning of the phase shift accumulated through superconducting loop  122  in qubit  100  can be accomplished by the application of a magnetic field or adjustment of the transport current I T  in superconducting loop  122 . 
     Additionally, embodiments of the present invention include at least one terminal junction  120 . Terminal junction  120  joins at least two terminals, terminals  110 - 1  through  110 -N. In some embodiments, the physical size of junction  120  is much less than the size of superconduction loop  122 . 
     Four-terminal SQUID devices are discussed in A. N. Omelyanchouk and Malek Zareyan, “Ballistic Four-Terminal Josephson Junction: Bistable States and Magnetic Flux Transfer”, Los Alamos preprint cond-mat/9905139, and B. J. Vleeming, “The Four-Terminal SQUID”, Ph.D. Dissertation, Leiden University, The Netherlands, 1998, both of which are herein incorporated by reference in their entirety. Four terminal SQUID devices are further discussed in R. de Bruyn Ouboter and A. N. Omelyanchouk, “Macroscopic Quantum Interference Effects in Superconducting Multiterminal Structures”,  Superlattices and Microstructures , Vol. 25 No 5/6 (1999), herein incorporated by reference in its entirety. 
     A quantum computation relies on qubit  100  including a qubit quantum system formed by supercurrents on superconducting loop  122  having degenerate ground states, designated |0&gt; and |1&gt;, of the supercurrent. An array of multi-terminal superconducting loops  122  having phase shifters  123  can be fabricated in useful numbers in a solid state structure. The ground state of the qubit quantum system includes two states that correspond to supercurrent flows that circulate clockwise and counterclockwise, respectively, in the plane of superconducting loop  122 . The qubit quantum system of qubit  100  can be initialized by the introduction of supercurrents from an external source some of terminals  110 - 1  through  110 -N not coupled to superconducting loop  122 . The ground-state of the qubit quantum system in each superconducting loop  122  containing phase shifter  123  is doubly degenerate in the absence of externally applied magnetic fields and/or transport currents I T  (each circulation direction has the same energy) and provides the basis for a qubit  100  for quantum computing in accordance with embodiments of the present invention. 
     The two degenerate states, corresponding to classical bit states, represented symbolically as |0&gt; and |1&gt;, are then the two basis states of the qubit quantum system of qubit  100 . The magnitude of the flux threading superconducting loop  122  can be much less than half a flux quantum Φ 0 , both because of intrinsic phase shifter  123  and the presence of the terminals  110 - 1  through  110 -N, which also introduces a phase. shift. At least two external terminals, terminals  110 - 2  and  110 - j  in  FIG. 13 , for example, of qubit  100  can be coupled to sources of transport current I 1 , and I j , respectively, in FIG.  13 . Terminals  110 - i  and  110 - j  along with the current source creates a transport current loop  127 . Additionally, an external magnetic flux, indicated by field {right arrow over (B)}, can be applied through superconducting loop  122  in order to control the physical parameters of the qubit quantum system of qubit  100 . By changing the transport current I T  and/or applying an external magnetic field {right arrow over (B)}, the magnitude of the flux threading superconducting loop  122 , the potential barrier between the two basis states |0&gt; and |1&gt; of the qubit quantum system of qubit  100 , and the tunneling matrix element Δ T (I) between the basis states of the qubit quantum system can be adjusted. 
     The choice of the physical sizes of the constriction junctions of junction  120 , tunnel junctions and/or semiconducting two-dimensional electron gas structures that couple the terminals, also affects the functioning of qubit  100 . To achieve a small total flux in superconducting loop  122  (which is desirable for decreasing the decoherence rate) and maximum influence of the transport current I T  on the properties of superconducting loop  122 , in some embodiments the links in the transport loop (e.g., the current loop providing current to junction  120 ) are much wider than the ones in superconducting loop  122 . A small residual flux exists because of spontaneous supercurrents, even in the absence of external fields. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system of qubit  100  will be affected most pronouncedly by the transport currents I T . 
     Multi-terminal junction  120  includes two important regimes: symmetric and asymmetric. Further, multi-terminal junction  120  can display symmetric, asymmetric, or a combination of symmetric and asymmetric properties. In a symmetric junction, a change in the direction of the transport current in the junction equally affects current in the terminals of the superconducting loop, thus having no overall affect on the current in the loop. Whereas, in an asymmetric junction, a change in the direction of the transport current differentially affects the terminals that form the superconducting loop, thus changing the overall current in the loop. 
     A symmetric junction can be used to reduce the potential energy barrier between the two nearly degenerate ground states of qubit  100 , thus providing a means of applying σ x  quantum gate operation. If a change in the direction of the transport current in terminals  110 - 2  through  110 - j  has an equal effect on the current in terminals  110 - 1  and  110 -N, then junction  120  is symmetric with respect to terminals  110 - 2  and  110 - j . Qubit  100 , under these circumstances, can then be referred to as a “symmetric qubit”. 
     An asymmetric junction can be used to bias one of the two ground states of the qubit, thus providing a means of applying σ z  quantum gate operation. If a change in the direction of the transport current in terminals  110 - 2  through  110 - j  causes a differential change in the current in the loop terminals  110 - 1  and  110 -N, then the junction is said to be asymmetric. Qubit  100 , then, can be referred to as an “asymmetric qubit”. 
     Therefore, a four-terminal junction  120  can be either a symmetric or asymmetric junction, and four-terminal qubit  100  can then be either a symmetric or asymmetric qubit. If qubit  100  includes a junction  120  with more than four terminals, for example a five-terminal qubit, then both symmetric and asymmetric properties can be realized. 
     An asymmetric qubit can be written to (i.e., the quantum states initialized) by applying a transport current I T  when the magnitude of the transport current I T  is larger than a threshold value which is determined by the specific implementation of qubit  100 . The direction of transport current I T  is chosen depending on which basis state (i.e., |0&gt; or |1&gt;) is being written into the qubit quantum system. The application of transport current I T , then, has the effect of biasing the qubit quantum system states into one of the degenerate basis states. In the biased state the qubit quantum system will decay to the most energetically favorable state (either |0&gt; or |1&gt; as required). In such systems, the time to decay typically is shorter than about 1 millisecond, depending on the particular embodiment of qubit  100 . Depending on the particular embodiment of qubit  100 , a magnetic field {right arrow over (B)} can also be applied in one of two directions, which can alter the time to decay. The magnetic field can be applied opposing the transport current induced bias, thereby decreasing the time for decay, or supporting the transport current induced bias, thereby increasing the time to decay. 
     A symmetric qubit can be written to (i.e., biased) by applying a static magnetic field {right arrow over (B)}. As described above, this will cause the qubit quantum system of qubit  100  to decay into the energetically favorable state on a time-scale dependent upon the embodiment of qubit  100  and the magnitude of an externally applied magnetic field {right arrow over (B)}. 
     Single qubit operations on asymmetric qubits can be performed by modulating the transport current and/or the external magnetic field strength. Setting the transport current I T  to zero sets the effective Hamiltonian describing the quantum system of qubit  100  proportional to {circumflex over (σ)} x , which is referred to as a Pauli matrix. In the basis where the qubit basis states |0&gt; and |1&gt; are chosen so that the state |0&gt; corresponds to the vector (1, 0) and the state |1&gt; corresponds to the vector (0, 1), 
           σ   ^     x     =       [         0       1           1       0         ]     .         
 
     This basis can be called the Z-diagonal basis. In this basis the Pauli matrix {circumflex over (σ)} x  rotates one of the basis states into the other basis state (i.e., {circumflex over (σ)} x |0&gt;=|1&gt; and {circumflex over (σ)} x |1&gt;=|0&gt;). 
     Increasing the transport current I T  past a threshold current, which can be an implementation dependent critical value, sets the Hamiltonian proportional to {circumflex over (σ)} z , which is another Pauli matrix. The matrix {circumflex over (σ)} z  is defined in the Z-diagonal basis to be 
           σ   ^     z     =       [         1       0           0         -   1           ]     .         
 
     The Pauli matrix {circumflex over (σ)} z  biases the states, or in an alternative interpretation adds a phase to the second state (i.e. {circumflex over (σ)} z |0&gt;=|0&gt; and {circumflex over (σ)} z |1&gt;=−|1&gt;). An arbitrary single qubit operation can be performed by performing combinations of the functions described by {circumflex over (σ)} x  and {circumflex over (σ)} z . 
     To keep the quantum system of qubit  100  in some specific state an alternating transport current I T (t) can be applied. The Hamiltonian representing the quantum system of qubit  100 , then, is proportional to I T (t){circumflex over (σ)} z . In some embodiments, for example, I T (t) can be a square wave. This method can be used in conjunction with a clock whose frequency is an integer multiple of the frequency of I T (t) so that, at every clock pulse, the quantum system of qubit  100  is in the same state in which it began. In addition, during the evolution of the qubit states external magnetic fields {right arrow over (B)} may be applied in accordance with a specific usage of the qubit. 
     Single qubit operations on symmetric embodiments of qubit  100  can be performed by modulating the transport current I T (t) and/or the external magnetic field {right arrow over (B)}. In the symmetric qubit embodiments of qubit  100 , changing the transport current I T (t) does not affect the bias (i.e., the initial quantum states of the qubit quantum system) but does affect the energy barrier between the qubit quantum system basis states. The effective Hamiltonian describing the quantum system of qubit  100 , then, includes a term proportional to Δ T (I){circumflex over (σ)} x  where the tunneling matrix element Δ T (I) can be varied over a large range dependent on the transport current I T (t). Applying a magnetic field {right arrow over (B)} normal to the plane of the superconducting loop  122  provides another term of the Hamiltonian that is proportional to {circumflex over (σ)} z . To keep the quantum system in some specific state, an alternating magnetic field normal to the superconducting loop  122 , {right arrow over (B)}(t), can be applied, adding a term proportional to {right arrow over (B)}(t){circumflex over (σ)} z  to the Hamiltonian. In some embodiments, for example, {right arrow over (B)}(t) can be modulated in a square wave. This method can also be used in conjunction with a clock whose frequency is an integer multiple of the frequency of {right arrow over (B)}(t) so that at every clock pulse the qubit quantum system of qubit  100  is in the same state in which it began. 
     In some embodiments, the state of the qubit quantum system of an asymmetric embodiment of qubit  100  begins with the application of an alternating transport current I T (t) as described above. The magnetic flux threading the superconducting loop  122  can then be measured using a suitable magnetic field measuring device, for example a SQUID magnetometer or a magnetic force microscope. The alternating transport current I T (t) causes the decay rate out of the desired final state to be minimized. Once the state of the qubit quantum system of qubit  100  is measured, the transport current I T (t) can be set to bias the states such that the measured state has a lower energy level, making the measured state&#39;s information stable so that it may be accessed at any later time. 
     In some embodiments, the state of the qubit quantum system of the asymmetric qubit can be measured by applying a transport current I T (t) in a fixed direction and then monitoring the voltage drop across pairs of external leads, for example  110 - 2 ,  110 - i  and  110 - j  of FIG.  13 . If the measured voltage remains constant then the state of the qubit quantum system corresponds to the favored energy state indicated by the fixed direction of the transport current I T (t). If the measured voltage changes, then the state of the qubit quantum system corresponds to an excited state indicated by the biasing of the fixed direction of the transport current I T (t). The changing voltage indicates a decay of the qubit quantum system state from the excited state. This indicates that the qubit quantum system was in the excited state relative to the bias indicated by the fixed direction of the transport current at the end of the calculation. 
     In some embodiments, the state of the quantum system of a symmetric qubit can be read similarly to the state of the quantum system of an asymmetric qubit, except that biasing is performed by application of magnetic fields normal to superconducting loop  122  and not via application of transport currents. 
       FIG. 1A  shows a plan view of an embodiment of a four-terminal constriction junction  120  according to the present invention. Four-terminal junction  120  includes terminals  110 - 1 ,  110 - 2 ,  110 - 3 , and  110 - 4  coupled at a constriction junction  140 . Superconducting currents I 1 , I 2 , I 3  and I 4  can exist in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. The terminal lengths L 1  through L 8 , which describe the linear dimensions of terminals  110 - 1 ,  110 - 2 ,  110 - 3 , and  110 - 4  as indicated in  FIG. 1A , can all be different and are typically chosen to be less than about 10 microns. The terminal widths W 1  through W 4 , which describe the widths of terminals  110 - 1  through  110 - 4 , respectively, can also all be different and are typically chosen to be less than the coherence length of the superconducting material of terminals  110 - 1  through  110 - 4 . For example, if terminals  110 - 1  through  110 - 4  are of aluminum, the coherence length is 1.6 microns. Four-terminal constriction junction  120  can be fabricated of any superconducting material. 
     In an exemplary embodiment, four-terminal constriction junction  120  can be fabricated of aluminum. Widths W 1  and W 2  can each be approximately 0.5 microns; widths W 3  and W 4  can each be approximately 0.05 microns; lengths L 1 , L 4 , L 5 , L 6 , L7, and L 8  can each be approximately 1 micron; and lengths L 2  and L 3  can each be approximately 0.55 microns. 
     In another exemplary embodiment, four-terminal constriction junction  120  can be fabricated of aluminum, with width W 1  approximately 0.5 microns, W 2  approximately 0.3 microns, W 3  approximately 0.08 microns, and W 4  approximately 0.05 microns, lengths L 1 , L 4 , L 5 , L 6 , and L 7  each approximately 1 micron, L 2  approximately 0.75 microns, L 3  approximately 0.68 microns, and L 8  approximately 0.9 microns. 
       FIG. 1B  shows a plan view of another embodiment of a four-terminal junction  120  with a constriction junction  142  coupling terminals  110 - 3  and  110 - 4  and two tunnel junctions  140  and  141 . Four-terminal junction  120  of  FIG. 1B  further includes terminals  110 - 1  and  110 - 2 . Tunnel junction  140  couples terminal  110 - 4  with terminal  110 - 1 . Tunnel junction  141  couples terminal  110 - 3  with terminal  110 - 2 . Superconducting currents I 1 , I 2 , I 3  and I 4  can exist in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. The terminal lengths L 1 -L 9 , which describe the linear dimensions of terminals  110 - 1  through  110 - 4  and the separation between junctions  140  and  141 , can all be different and are typically chosen to be less than about 10 microns. The terminal widths W 1 -W 4 , which as before describe the widths of terminals  110 - 1  through  110 - 4 , respectively, can also be different and are typically chosen to be less than the coherence length of the superconductor used. For example, in the case of aluminum, the coherence length is about 1.6 microns. Four-terminal junction  120  can be fabricated of any superconductor material. Tunnel junction  140  is typically fabricated using an insulating layer between terminals  110 - 4  and  110 - 1 . Tunnel junction  141  is typically fabricated using an insulating layer between terminals  110 - 2  and  110 - 3 . 
       FIG. 1C  shows a cross-sectional diagram of the embodiment of four-terminal junction  120  shown in FIG.  1 B. Terminals  110 - 4  and  110 - 1  are separated by insulator  150  at junction  140  and terminals  110 - 3  and  110 - 2  are separated by insulator  150  at junction  141 . 
     In an exemplary embodiment, four-terminal junction  120  as shown in  FIG. 1B  can be fabricated of aluminum, with widths W 1  and W 2  being approximately 0.05 microns, widths W 3  and W 4  being approximately 0.5 microns, lengths L 1 , L 2 , L 3 , L 4 , L 5 , and L 7  being approximately 1 micron, and L 9  being approximately 0.1 microns. 
     Tunnel junction  140  can be fabricated where insulation layer  150  ( FIG. 1C ) is an aluminum oxide layer of thickness approximately 0.05 microns. Insulating layer  150  can be deposited onto terminal  110 - 4  Terminal  110 - 1  is then deposited on insulating layer  150 . The tunnel junction  141  can also be fabricated where insulating layer  150  ( FIG. 1C ) is an aluminum oxide layer of thickness approximately 0.05 microns which is deposited onto terminal  110 - 3 , upon which is deposited terminal  110 - 2 . Although in  FIG. 1C , terminals  110 - 1  and  110 - 2  are shown above terminals  110 - 4  and  110 - 3 , respectively, in general terminals  110 - 1  and  110 - 2  can be on either side of terminals  110 - 4  and  110 - 3 , respectively. 
     In another exemplary embodiment, four-terminal junction  120  as shown in  FIGS. 1B and 1C  can be fabricated of aluminum with the following dimensions: width W 1  being approximately 0.05 microns; width W 2  being approximately 0.08 microns; width W 3  being approximately 0.3 microns; width W 4  being approximately 0.5 microns; lengths L 2 , L 3 , L 4 , L 5 , and L 7  being approximately 1 micron; length L 1  being approximately 0.8 microns; and length L 9  being approximately 0.1 microns. Again, junctions  140  and  141  can be fabricated with insulation layer  150  being an approximately 0.05 microns layer of aluminum oxide deposited onto terminals  110 - 4  and  110 - 3 , respectively. Terminals  110 - 1  and  110 - 2  can then be deposited at junctions  140  and  141  on insulating layer  150 . 
       FIG. 1D  shows a plan view of another embodiment of four-terminal junction  120  with one constriction junction  142  coupling terminals  110 - 3  and  110 - 4  and tunnel junctions  140  and  141 , which are fabricated so as to run parallel to terminals  110 - 4  and  110 - 3  over distances D 1  and D 2 , respectively. In some embodiments, D 1  and D 2  are each greater than approximately 0.2 microns. Tunnel junction  140  couples terminal  110 - 4  with terminal  110 - 1 . Tunnel junction  141  couples terminal  110 - 2  with terminal  110 - 3 . Superconducting currents I 1 , I 2 , I 3  and I 4  are the currents in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. The terminal lengths L 2 , L 4 , L 5  and L 7  can all be different and are typically chosen to be less than about 10 microns. Note that in some embodiments the length L 9  can be negative, which may result in overlapping tunnel junctions  140  and  141 . Terminals  110 - 1  and  110 - 2  can extend at arbitrary angles A 120-1  and A 120-2  respectively. The terminal widths W 1 -W 4  can all be different and are typically chosen to be less than the coherence length of the superconducting material used. For example, in the case of aluminum terminals, the coherence length is about 1.6 microns. Four-terminal junction  120  can, however, be fabricated of any superconducting material. As shown in  FIG. 1E , tunnel junction  140  can be fabricated using an insulating layer  150  between terminals  110 - 4  and  110 - 1 . Tunnel junction  141  can be fabricated using an insulating layer  151  between terminals  110 - 3  and  110 - 2 . 
     In an exemplary embodiment, four-terminal junction  120  as shown in  FIGS. 1D and 1E  can be fabricated of aluminum with the following dimensions: widths W 1  and W 2  being approximately 0.05 microns; widths W 3  and W 4  being approximately 0.5 microns; lengths L 2  and L 4  being approximately 1 micron; lengths D 1  and D 2  being approximately 1 micron; lengths L 5  and L 7  being approximately 1.5 microns; and L 9  being approximately 0.1 microns. Tunnel junction  140  can be fabricated with insulating layer  150  being an aluminum oxide layer of thickness approximately 0.05 microns deposited onto terminal  110 - 4 , upon which is deposited terminal  110 - 1 . Tunnel junction  141  can be fabricated with insulating layer  151  being an aluminum oxide layer of thickness approximately 0.05 microns which is deposited onto terminal  110 - 3 , upon which is deposited terminal  110 - 2 . 
     In another exemplary embodiment of junction  120  as shown in  FIGS. 1D and 1E , four-terminal junction can be fabricated of aluminum with the following dimensions: width W 1  being approximately 0.05 microns; width W 2  being approximately 0.08 microns; width W 3  being approximately 0.3 microns; width W 4  being approximately 0.5 microns; lengths L 2  and L 4  being approximately 1 micron; lengths D1 and D2 being approximately 1 micron; lengths L 5  and L 7  being approximately 1.5 microns; and length L 9  being approximately 0.1 microns. Again, insulating layer  150  forming part of tunnel junction  140  can be fabricated of aluminum oxide with thickness approximately 0.05 microns and insulating layer  151  forming part of tunnel junction  141  can be fabricated of aluminum oxide of thickness approximately 0.05 microns. 
       FIG. 1F  shows a plan view of an embodiment of a four-terminal junction  120  with constriction junction  141  coupling terminals  110 - 2 ,  110 - 3  and  110 - 4  and tunnel junction  140  coupling junctions  110 - 4  with  110 - 1 . Superconducting currents I 1 , I 2 , I 3  and I 4  can exist in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. The terminal lengths L 1 , L 2 , L 3 , L 4 , L 5  and L 7 , which indicate the dimensions of terminals  110 - 1  through  110 - 4 , can all be different and are typically chosen to be less than about 10 microns. The widths of terminals  110 - 1  through  110 - 4 , W 1 -W 4  respectively, can also be different and are typically chosen to be less than the coherence length of the superconductor used. Junction  120  can be fabricated of any superconducting material. Tunnel junction  140  can typically be fabricated using an insulating layer  150  between terminals  110 - 4  and  110 - 1 , as was shown for junction  140  in FIG.  1 C. 
     In an exemplary embodiment, four-terminal junction  120  as shown in  FIG. 1F  can be fabricated of aluminum with the following dimensions: widths W 1 , and W 4  being approximately 0.5 microns; widths W 2  and W 3  being approximately 0.05 microns; lengths L 1 , L 2 , L 3 , L 4 , L 5 , and L 7  being approximately 1 micron; and length L 9  being approximately 0.1 microns. Tunnel junction  140  can be fabricated with insulating layer  150  being an aluminum oxide layer of thickness approximately 0.05 microns. 
     In another exemplary embodiment of junction  120  as shown in  FIG. 1F , four-terminal junction can be fabricated of aluminum with the following dimensions: width W 1  being approximately 0.05 microns; width W 4  being approximately 0.08 microns; width W 2  being approximately 0.3 microns; width W 3  being approximately 0.5 microns; lengths L 1 , L 2 , L 3 , L 4 , L 5 , and L 7  being approximately 1 micron; and length L 9  being approximately 0.1 microns. Tunnel junction  140  can, as before, include insulating layer  150  fabricated using an aluminum oxide layer of thickness approximately 0.05 microns. 
       FIG. 1G  shows a plan view of another embodiment of a four-terminal junction  120  having constriction junction  141  coupling terminals  110 - 2 ,  110 - 3  and  110 - 4  and tunnel junction  140 , which is fabricated so as to run parallel to terminal  110 - 4  at least over distance D 1 , coupling terminals  110 - 4  with  110 - 1 . In some embodiments, distance D 2  is greater than approximately 0.2 microns. As before, terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4  can carry superconducting currents I 1 , I 2 , I 3  and I 4 , respectively. Terminal lengths L 2 , L 3 , L 4 , L 5  and L 7  can all be different and are typically chosen to be less than about 10 microns. The terminal widths W 1 -W 4  of terminals  110 - 1  through  110 - 4 , respectively, can all be different and are typically chosen to be less than the coherence length of the superconducting material used. Four-terminal junction  120  can be fabricated of any superconductor. As discussed before, tunnel junction  140  is typically fabricated with an insulating layer  150  between terminals  110 - 4  and  110 - 1 , as shown in FIG.  1 E. 
     In an exemplary embodiment of junction  120  as shown in  FIG. 1G , four-terminal junction can be fabricated of aluminum with the following dimensions: widths W 1  and W 4  each being approximately 0.05 microns; widths W 2  and W 3  each being approximately 0.5 microns; lengths L 2 , L 3 , L 4 , and L 5  each being approximately 1 micron; length D 1  being approximately 0.5 microns; length L 9  being approximately 0.1 microns; and length L 7  being approximately 1.5 microns. Tunnel junction  140  can include insulating layer  150  as shown in  FIG. 1E  which can be fabricated using an aluminum oxide layer of thickness approximately 0.05. 
     In another exemplary embodiment of junction  120  as shown in  FIG. 1G , four-terminal junction can be fabricated of aluminum with the following dimensions: width W 1  being approximately 0.05 microns; width W 4  being approximately 0.08 microns; width W 2  being approximately 0.3 microns; width W 3  being approximately 0.5 microns; lengths L 2 , L 3 , L 4 , and L 5  being approximately 1 micron, length D 1  being approximately 0.5 microns, length L 9  being approximately 0.1 microns; and length L 7  being approximately 1.5 microns. Again, tunnel junction  140  can include insulating layer  150  which can be fabricated using an aluminum oxide layer of thickness approximately 0.05 microns. 
       FIG. 1H  shows a plan view of an embodiment of a four-terminal junction  120  where the four terminals  110 - 1 ,  110 - 2 ,  110 - 3 , and  110 - 4  are all coupled by junction  140 , which in this embodiment is a two dimensional electron gas structure. Again, terminals  110 - 1 ,  110 - 2 ,  110 - 3 , and  110 - 4  can carry superconducting currents I 1 , I 2 , I 3  and I 4 , respectively. The terminal lengths L 1 -L 8  can all be different and are typically chosen to be less than about 10 microns. The terminal widths W 1 -W 4  also can all be different and are typically chosen to be less than the coherence length of the superconductor used. Superconducting terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4  can be fabricated of any superconducting material which can be coupled to a two dimensional electron gas structure of junction  140 . The two dimensional electron gas structure of junction  140  can be fabricated of any structure which allows electrons to be confined to a single two dimensional plane and allows coupling of these electrons to superconducting terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 . 
     An exemplary embodiment of four-terminal junction  120  as shown in  FIG. 1H  can be fabricated of niobium with the following dimensions: widths W 1  and W 2  being approximately 0.2 microns; widths W 3  and W 4  being approximately 0.05 microns; and lengths L 1 , L 2 , L 3 , L 4 , L 5 , L 6 , L 7 , and L 8  each being approximately 0.5 microns. The two dimensional electron gas structure of junction  140  can be constructed of a semiconducting InAs heterostructure. The details of the fabrication and behavior of two dimensional electron gas junctions are well known, see e.g., A. Jacobs, R. Kümmel, and H. Plehn, “Proximity Effect, Andreev Reflections, and Charge Transport in Mesoscopic Superconducting-Semiconducting Heterostructures,” Los Alamos Preprint cond.-mat/9810343 v2 (1998), republished in  Superlattices and Microstructures , Vol. 25, Nr. 5/6, 669-681 (1999), which is herein incorporated by reference in its entirety. 
     Another exemplary embodiment of four-terminal junction  120  as shown in  FIG. 1H  can be fabricated of niobium with the following dimensions: width W 1  being approximately 0.1 microns; width W 2  being approximately 0.08 microns; width W 3  being approximately 0.03 microns; width W 4  being approximately 0.01 microns; and lengths L 1 , L 2 , L 3 , L 4 , L 5 , L 6 , L 7 , and L 8  each being approximately 0.5 microns. The two dimensional electron gas structure of junction  140  can be constructed of semiconducting InAs. 
       FIG. 1I  shows a plan view of another embodiment of a four-terminal junction  120 , with constriction junction  141  coupling terminals  110 - 1  and  110 - 2  and junction  140 , which can be a two dimensional electron gas structure, coupling terminals  110 - 4  and  110 - 3  with constriction junction  141 . Terminals  110 - 1 ,  110 - 2 ,  110 - 3 , and  110 - 4  can carry superconducting currents I 1 , I 2 , I 3  and I 4 , respectively. As before, all of the terminal lengths (L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , L 8  and L 9 ) indicating the dimensions of terminals  110 - 1  through  110 - 4  can all be different and are typically chosen to be less than about 10 microns. The widths of terminals  110 - 1  through  110 - 4 , W 1 -W 4 , can all be different and are typically chosen to be less than the coherence length of the superconducting material utilized in their fabrication. Terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4  can be fabricated of any superconducting material which can be coupled to a two dimensional electron gas junction  140 . 
     In an exemplary embodiment, four-terminal junction  120  as shown in  FIG. 1I  can be fabricated of niobium with the following dimensions: widths W 1  and W 2  being approximately 0.5 microns; widths W 3  and W 4  being approximately 0.05 microns; lengths L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , L 8 , each being approximately 0.5 microns; and length L 9  being approximately 0.1 microns. The two dimensional electron gas structure of junction  140  can be formed of InAs. 
     In another exemplary embodiment, four-terminal junction  120  can be fabricated of niobium with the following dimensions: width W 1  being approximately 0.05 microns; width W 4  being approximately 0.08 microns; width W 2  being approximately 0.01 microns; width W 3  being approximately 0.15 microns; lengths L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , and L 8  being approximately 0.5 microns; and length L 9  being approximately 0.1 microns. The two dimensional electron gas structure of junction  140  can be formed of InAs. 
       FIG. 1J  shows a plan view of an embodiment of a four-terminal junction  120  having tunnel junction  141  coupling terminals  110 - 1  and  110 - 2  and two dimensional electron gas structure junction  140  coupling terminals  110 - 3  and  110 - 4  with junction  141 . Superconducting currents I 1 , I 2 , I 3  and I 4  can exist in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. The terminal lengths indicating the dimensions of terminals  110 - 1  through  110 - 4 , L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , L 8 , and D 1  in  FIG. 1J , can all be different and are typically chosen to be less than 10 microns. The terminal widths W 1 -W 4 , indicating the widths of terminals  110 - 1  through  110 - 4 , can all be different and are typically chosen to be less than the coherence length of the superconducting material of terminals  110 - 1  through  110 - 4 . Terminals  110 - 1  through  110 - 4  can be fabricated of any superconducting material which can be coupled to the two dimensional electron gas structure of junction  140 . Tunnel junction  141  is typically fabricated by introducing a layer of insulating material, such as aluminum oxide, between terminals  110 - 1  and  110 - 2 , as was previously discussed with respect to  FIGS. 1C and 1E . 
     An exemplary embodiment of four-terminal junction  120  can be fabricated of niobium with the following dimensions: widths W 1  and W 2  being approximately 0.5 microns; widths W 3  and W 4  being approximately 0.05 microns; lengths L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , and L 8  each being approximately 0.5 microns; and length D 1  being approximately 0.1 microns. As previously discussed, the two dimensional electron gas structure of junction  140  can be formed of InAs. Additionally, tunnel junction  141  can be fabricated using a niobium oxide layer of thickness approximately 0.05 microns which is deposited onto terminal  110 - 2 , upon which is deposited terminal  110 - 1 . 
     Another exemplary embodiment of four-terminal junction  120  can be fabricated of niobium with the following dimensions: width W 1  being approximately 0.05 microns; width W 4  being approximately 0.08 microns; width W 2  being approximately 0.05 microns; width W 3  being approximately 0.15 microns; lengths L 1 , L 2 , L 4 , L 5 , L 6 , L 7 , and L 8  each being approximately 0.5 microns; and length D 1  being approximately 0.1 microns. The two dimensional electron gas structure of junction  140  can be formed of InAs and tunnel junction  141  can be formed with junctions  110 - 1  and  110 - 2  separated by a niobium oxide layer of approximately 0.05 microns thickness. 
       FIG. 1K  shows a plan view of an embodiment of a multi-terminal junction  120  with constriction junction  140  coupling terminals  110 - 1  through  110 -N, where N is an arbitrary integer. Terminals  110 - 1  through  110 -N can carry superconducting currents I 1  through I N , respectively. The terminal lengths L 1  through L 2N , which describe the dimensions of terminals  110 - 1  through  110 -N, can all be different and are typically chosen to be less than about 10 microns. The widths of terminals  110 - 1  through  110 -N, W 1 -W N , can also all be different and are typically chosen to be less than the coherence length of the superconducting material of multi-terminal junction  120 . Multi-terminal constriction junction  120  can be fabricated of any superconducting material. In accordance with aspects of the present invention, multi-terminal junction  120  can be an element in superconducting circuits. 
       FIG. 1L  shows a plan view of an embodiment of a multi-terminal junction  120  having a constriction junction  140 - 2  coupling terminals  110 -N and  110 -(N−1), a tunnel junction  140 - 1  coupling terminals  110 -N with  110 - 1 , and tunnel junction  140 - 3  through  140 -(N−1) coupling junctions  110 - 2  through  110 -(N−2), respectively, to terminal  110 -(N−1), where N is an arbitrary number. Terminals  110 - 1  through  110 -N can carry superconducting currents I 1  through I N , respectively. The lengths L 1  through L 2N , which characterize the dimensions of terminals  110 - 1  through  110 -N, can all be different and are typically chosen to be less than about 10 microns. The widths of terminals  110 - 1  through  110 -N, W 1 -W N , can all be different and are typically chosen to be less than the coherence length of the superconductor used. The separation between junctions, D 1  through D N−2 , can also all be different and are typically chosen to be less than about 10 microns. Multi-terminal junction  120  can be fabricated of any superconducting material. In accordance with certain aspects of the present invention, multi-terminal junction  120  can be utilized as a circuit element in superconducting circuits. 
       FIG. 1M  shows a plan view of another embodiment of multi-terminal junction  120 . Multi-terminal junction  120  of  FIG. 1M  includes a two dimensional electron gas junction  140  coupling terminals  110 - 1  through  110 -N, where N is an integer. As before, terminals  110 - 1  through  110 -N can carry superconducting currents I 1  through I N , respectively. The terminal lengths L 1  through L (2N) , which indicate the dimensions of terminals  110 - 1  through  110 -N, can all be different and are typically chosen to be less than about 10 microns. The widths of terminals  110 - 1  through  110 -N, W 1 -W N , can all be different and are typically chosen to be less than the coherence length of the superconducting material of terminals  110 - 1  through  110 -N. Terminals  110 - 1  through  110 -N can be fabricated from any superconducting material. Two dimensional electron gas junction  140  can, for example, be formed of InAs. In an exemplary embodiment the two dimensional electron gas is formed of an InAs layer deposited on an AlSb substrate. Terminals  110 - 1  through  110 -N can be formed of niobium and can be deposited on the InAs layer. In accordance with certain embodiments of the invention, multi-terminal junction  120  can be used as a circuit element in superconducting circuits. 
     In general, multi-terminal junction  120  can include any number of terminals coupled by any types of junction. The embodiments of junction  120  shown in  FIGS. 1A through 1M  are exemplary only. One skilled in the art will recognize a multitude of variations from those shown above. Those variations are intended to be within the scope of this disclosure. 
     Additionally, qubit  100  shown in  FIG. 13  illustratively shows terminals  110 -N and  110 - 1  coupled to portions  124  and  125  of superconducting loop  122 . In general, superconducting loop  122  can be formed with any arbitrary pair of terminals  110 - 1  through  110 -N. Further, qubit  100  shows phase shifter  123  as part of superconducting loop  122 . 
       FIG. 2A  shows a plan view of an embodiment of a two terminal phase shifter  123  having a S/N/D/N/S heterostructure. Phase shifter  123  of  FIG. 2A  includes an s-wave superconducting terminal  210  coupled to a normal metal connector  250  which is coupled to a d-wave superconductor  240  which, in turn, is coupled to a normal metal connector  251  which is coupled to an s-wave superconducting terminal  211 . In some embodiments all lengths and widths L S0 , L S1 , L S2 , L S3 , W S0 , and W S1 , indicating the termination of terminals  210  and  211 , can all be different. In some embodiments, the terminal lengths and widths can all be less than about five microns. 
     Modifying the angle of contact between the d-wave superconductor  240  and each of the two external terminals  210  and  211  modifies the phase shift acquired in transit through phase shifter  123  in a known way. For example, if the normal metal connectors,  250  and  251 , are at a right angle to each other, the total phase is shifted by π through the phase shifter  123 . Furthermore, if the normal metal connectors were directly opposite (0° apart), then there would be no accumulated phase shift. Following from this, any angle between 0° and 90°, where the angle is represented by θ, would lead to a phase shift of 2θ.  FIG. 2B  illustrates an exemplary embodiment of a π-phase shifter. The angle θ is 90° and the normal metal connector  250  is directly parallel with the crystal alignment of the junction  240 . There is no restriction that the normal metal connector  250  be directly parallel with the crystal alignment of the junction  240 . 
     The physical characteristics of the normal metal connectors  250  and  251  can be chosen so as to provide standard Josephson junction connections between terminal  210  and d-wave superconductor  240  and terminal  211  and d-wave superconductor  240 , respectively. Currents flowing in terminals  210  and  211  are labeled I S0  and I S1 , respectively. The dimensions of d-wave superconductor  240  and connectors  250  and  251  are not critical. 
     In some embodiments terminals  210  and  211  can be niobium (Nb), aluminum (Al), lead (Pb) or tin (Sn). In some embodiments d-wave superconductor  240  can be YBa 2 Cu 3 O 7-d , where 0.2&lt;d&lt;0.8. In accordance with an exemplary embodiment, terminals  210  and  211  can be made of niobium, connectors  250  and  251  of gold, and d-wave superconductor  240  of YBa 2 Cu 3 O 6.68 . Lengths L S0 , L S1 , L S2 , and L S3  can be approximately 0.5 microns, widths W S0  and W S1  can be approximately 0.5 microns, and connectors  250  and  251  can be approximately 0.05 microns thick. The embodiment of phase shifter  123  shown in  FIG. 2B  will produce a phase shift of π in the superconducting order parameter accumulated in transition between terminals  210  and  211 . 
       FIG. 2C  shows a plan view of another embodiment of two terminal phase shifter  123 . Phase shifter  123  as shown in  FIG. 2C  includes a heterostructure containing a grain boundary junction  260  between two lattice-mismatched d-wave superconductors  241  and  242 . In some embodiments the d-wave superconductors  241  and  242  can be YBa 2 Cu 3 O 7-d , where 0.2&lt;d&lt;0.8. Modifying the angle of mismatch of the d-wave order parameters at grain boundary  260  between superconductors  241  and  242  affects the phase shift across grain boundary  260  in a known way. For example,  FIG. 2C  illustrates a mismatch angle of 45° which would lead to a π/2-phase shift. The behavior and resulting phase shift of such junctions are well known and are described in detail in C. Bruder, A. van Otterlo, and G. T. Zimanyi, “Tunnel Junctions of Unconventional Superconductors,” Phys. Rev. B 51, 12904-07 (1995), and furthermore in R. R. Schultz, B. Chesca, B. Goetz, C. W. Schneider, A. Schmehl, H. Bielefeldt, H. Hilgenkamp, J. Mannhart, and C. C. Tsuei, “Design and Realization of an all d-Wave dc π-Superconducting Quantum Interference Device,”  Applied Physics Letters , 76, p. 912-14 (2000), each of which are herein incorporated by reference in their entirety. In some embodiments the mismatch at grain boundary  260  can be achieved by deposition of the d-wave superconductor onto a bi-crystal substrate with an existing lattice-mismatched grain boundary which the d-wave superconductor inherits. 
       FIG. 2D  shows a cross sectional view of phase shifter  123  as shown in FIG.  2 C. Superconductors  242  and  241  are grown on substrate crystals  271  and  270 , respectively. Substrate crystals  271  and  270  can be mounted on substrate  272 . In some embodiments bi-crystal substrate  270  and  271  can be an insulator such as SrTiO 3  (strontium titanate) or Ti:Al 2 O 3  (sapphire) which are commercially available. 
     In some embodiments, grain boundary  260  can be created by using a bi-epitaxial method where a d-wave superconductor is deposited onto a substrate containing seed layers upon which the d-wave superconductor grows in a different crystallographic direction than on the substrate itself. In some embodiments the substrate can be an insulator such as strontium titanate and the seed layers can be CeO (cerium oxide) or MgO (magnesium oxide). See F. Tafuri, F. Carillo, F. Lombardi, F. Miletto Granozio, F. Ricci, U. Scotti di Uccio, A. Barone, G. Testa, E. Sarnelli, J. R. Kirtley, “Feasibility of Biepitaxial YBa 2 Cu 3 O 7-x  Josephson Junctions for Fundamental Studies and Potential Circuit Implementation, Los Alamos preprint cond-mat/0010128, published  Phys. Rev. B  62, 14431-38 (2000), which is herein incorporated by reference in its entirety. 
     In some embodiments normal metal connector  250  couples d-wave superconductor  241  to s-wave superconducting terminal  211 . In some embodiments normal metal connector  251  couples d-wave superconductor  242  to s-wave superconducting terminal  210 . In some embodiments normal metal connectors  250  and  251  can be gold (Au), silver (Ag), platinum (Pt), or any other normal metal substance; and s-wave superconducting terminals  210  and  211  can be aluminum (Al), niobium (Nb), or any other superconductor with s-wave pairing symmetry. 
     In some embodiments all lengths and widths L S0 , L S1 , L S2 , L S3 , W S0 , and W S1  can all be different, in some embodiments each of the lengths can be less than about one micron. The physical characteristics of normal metal connectors  250  and  251  can be chosen so as to provide standard Josephson junction connections between terminals  210  and d-wave superconductor  241  and terminals  211  and d-wave superconductor  240 , respectively. Currents flowing in terminals  210  and  211  are labeled I S0  and I S1 , respectively. The dimensions of d-wave superconductor  240  and connectors  250  and  251  are not critical. 
     In accordance with an exemplary embodiment of phase shifter  123  as shown in  FIG. 2C , terminals  210  and  211  can be made of niobium, connectors  250  and  251  of gold, and d-wave superconductor  240  and  241  of YBa 2 Cu 3 O 6.68 . Lengths L S0 , L S1 , L S2 , and L S3  can be approximately 0.5 microns, widths W S0  and W S1  can be approximately 0.5 microns, and connectors  250  and  251  can be approximately 0.05 microns thick. The d-wave superconductors  240  and  241  can have a symmetric 22.5/22.5 degree lattice mismatch, in which the crystallographic a-axis of d-wave superconductor  240  makes an angle of +22.5 degrees with grain boundary  260  and the crystallographic a-axis of d-wave superconductor  241  makes an angle of −22.5 degrees with grain boundary  260 . This type of grain boundary is typically called a symmetric 45 degree grain boundary, as the angle between the crystallographic a-axes of superconductors  240  and  241  is 45 degrees. This embodiment will produce a phase shift of π in the superconducting order parameter accumulated in transition across grain boundary  260 . It is also “quiet” in the sense that no spontaneous supercurrents or magnetic fluxes are produced at a symmetric 45 degree grain boundary and therefore noise due to phase shifter  123  in a superconducting electronic circuit is reduced. 
       FIG. 2E  shows a plan view of another embodiment of a two terminal phase shifter  123 . Phase shifter  123  of  FIG. 2E  includes a junction between s-wave superconductor  210 , ferromagnetic region  276  and s-wave superconductor  211 . In this embodiment the s-wave superconductor/ferromagnet/s-wave superconductor junction is in the axis normal to the plane shown in FIG.  2 E.  FIG. 2F  shows a cross sectional view of phase shifter  123  with ferromagnet  275  between s-wave superconductor  210  (on the bottom) and s-wave superconductor  211  (on the top). An insulating barrier  275 , shown in gray in  FIG. 2E , provides insulation between terminals  210  and  211 . 
     Modifying the geometry of the ferromagnetic region  275  can change the angle of the phase shift in a known way. In  FIG. 2E , lengths L S1  and L S3 , as before, indicate the lengths of terminals  210  and  211 , respectively. H T0  and H T1  indicate the distance between the edge of terminals  210  and  211 , respectively, and the edge of insulation region  275 . The quantities H F  and W F  indicate the height and width of ferromagnetic region  276 , respectively. The quantity D T1  indicates the distance between the edge of terminal  211  and the edge of terminal  210 . In some embodiments, lengths and widths D T1 , H T1 , L S2 , H T0 , W S0 , and W S1  can be all different and, in some embodiments, all be less than about five microns. In some embodiments lengths H F  and W F  can be different and, in some embodiments, can be less than about one micron, with these lengths chosen so as to give a required phase shift. Currents flowing in terminals  210  and  211  are labeled I S0  and I S1 , respectively. 
     In some embodiments terminals  210  and  211  can be niobium (Nb), aluminum (Al), lead (Pb), tin (Sn), or any other superconductor with s-wave pairing symmetry. In some embodiments insulating region  275  can be aluminum oxide (AlO 2 ) or any other insulating material. In some embodiments ferromagnetic region  276  can be an alloy of copper and nickel (Cu:Ni) or any other ferromagnetic material. One method for fabricating an example of phase shifter  123  as shown in  FIGS. 2E and 2F  is described in V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Veretennikov, A. A. Golubov, J. Aarts, “Coupling of Two Superconductors Through a Ferromagnet: Evidence for a π-Junction,” Los Alamos preprint server cond-mat/0008364, submitted to  Phys. Rev. Lett . (2000), which is herein incorporated by reference in its entirety. 
       FIG. 2G  shows a plan view of another embodiment of a two terminal phase shifter  123  having a ferromagnetic region  276  imbedded in a junction between s-wave superconductors  210  and  211 . In this embodiment the s-wave superconductor/ferromagnet/s-wave superconductor junction is in the plane of FIG.  2 G. Thus, the ferromagnetic region,  276 , is directly in the plane with of the terminals  210  and  211 . Modifying the geometry of the ferromagnetic region  276  can change the angle of the phase shift in a known way. In some embodiments lengths and widths D T1 , HT T1 , L S2 , W S0 , and W S1  can be all different and, in some embodiments, all be less than about five microns. In some embodiments lengths H F  and W F  can be different and less than about one micron, with these lengths chosen so as to give a required phase shift. Currents flowing in terminals  210  and  211  are labeled I S0  and I S1 , respectively. In some embodiments terminals  210  and  211  can be niobium (Nb), aluminum (Al), lead (Pb) tin (Sn), or any other superconductor with s-wave pairing symmetry. In some embodiments ferromagnetic region  276  can be an alloy of copper and nickel (Cu:Ni) or any other ferromagnetic material. Ferromagnetic region  276  can be prepared by, for example, implantation of a ferromagnetic substance into a superconducting junction. 
       FIG. 3A  shows a plan view of an embodiment of a four-terminal qubit  100  having junction  120  and phase shifter  123  in superconducting loop  122 . Junction  120  can be any four-terminal junction. Examples of embodiments of junction  120  are shown in  FIGS. 1A through 1M . Examples of embodiments of phase shifter  123  are shown in  FIGS. 2A through 2G . In Qubit  100  as shown in  FIG. 3A , terminal  110 - 1  can be reduced from width W 1  to width W J1  before being coupled to junction  120 , terminal  110 - 2  can be reduced from width W 2  to width W J2  before being coupled to junction  120 , terminal  110 - 3  can be reduced from width W 3  to width W J3  before being coupled to junction  120 , and terminal  110 - 4  can be reduced from width W 4  to width W J4  before being coupled to junction  120 . Terminal  210  of phase shift  123  is coupled to portion  124  having width W S4  and terminal  211  of phase shift  123  is coupled to portion  125  having width W S6 . Terminal  110 - 3  is coupled to portion  125  having width W S5 . 
     Terminals  110 - 1 ,  110 - 2 ,  110 - 3 ,  110 - 4 ,  210 , and  211  along with portions  124  and  125  can be made of any superconducting material compatible with the particular embodiments of phase shifter  123  and four-terminal junction  120 . Superconducting currents I 1 , I 2 , I 3  and I 4  can exist in terminals  110 - 1 ,  110 - 2 ,  110 - 3  and  110 - 4 , respectively. Widths W J1  through W J4  are constrained only by the requirement of compatibility with junction  120  and widths W S0  and W S1  are compatible with phase shifter  123 . In some embodiments, widths W 1  through W 4 , W J1  through W J4 , and W S0  through W S6  can all be less than about 10 microns. Lengths L 1 , L 2 , D J1 , D J2 , D J3 , D J4 , D P0 , and D P1  are all compatible with phase shifter  123  and four-terminal junction  120  and, in some embodiments, can be all less than about 10 microns. Superconducting loop  122  can be threaded by a magnetic flux Φ, which may contain contributions from a spontaneous supercurrent in the loop and externally applied magnetic fields. 
     Four-terminal junction  120  couples one end of each of the four terminals  110 - 1  through  110 - 4  by, for example, constriction junctions, tunnel junctions, two-dimensional electron gas structures, or combinations of these. The choice of the physical sizes of the elements in four-terminal junction  120  that couple the four terminals  110 - 1  through  110 - 4  also affects the function of qubit  100 . To achieve a small magnetic flux Φ in superconducting loop  122  (which is desirable for coherence consideration) and maximum influence of the transport current I T =I 1 =I 2  on the properties of superconducting loop  122 , in some embodiments the links in the transport loop are much wider than the ones in superconducting loop  122 , that is, W J1  and W J2  are much larger than W J3  and W J4 . A small magnetic flux Φ can exist even in the absence of external magnetic fields because of spontaneous supercurrents. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system of qubit  100  will be affected most pronouncedly by the applied transport current I T =I 1 =I 2 . 
     SQUID loop  122  with intrinsic phase shifter  123  can provide a basic block for construction of qubit  100  but can also be utilized for demonstration of macroscopic quantum tunneling and incoherent quantum noise in a solid state system. As described further below, the macroscopic quantum tunneling in a set of independent four-terminal qubits  100  with intrinsic phase shifters  123  (i.e., with no entanglements between individual qubits) permits construction of a random number generator that generates random series with zero correlation between numbers in the random series. 
     Four-terminal qubit  100  with intrinsic phase shifter  123  includes a superconducting loop  122  linking two of terminals  110 - 1  through  110 - 4 , terminals  110 - 4  and  110 - 3  in FIG.  3 A. Four-terminal qubit  100  with intrinsic phase shifter  123  can be made superconducting by reducing the temperature of qubit  100  below the superconducting critical temperature T c  of all of the superconducting materials utilized in the formation of qubit  100 . Four-terminal junction  120  may be either symmetric or asymmetric, as was discussed with respect to  FIGS. 1A through 1M . The superconducting materials from which qubit  100  is constructed are constrained only by the requirement of compatibility with phase shifter  123  and four-terminal junction  120  and otherwise may have any pairing symmetry. For example, materials used may be s-wave, for example, niobium or aluminum, or d-wave, such as a high-Tc cuprate, for example YBa 2 Cu 3 O 7-x , or any superconducting material in which the Cooper pairs are in a state with non-zero orbital angular momentum. 
     Four-terminal qubit  100  with intrinsic phase shifter  123  can be formed on an insulating substrate such as, for example, strontium titanate or sapphire. (See  FIG. 2C , for example). Phase shifter  123  may be any structure that shifts the phase of the superconducting order parameter in transition across the structure. Examples of phase shifter  123  are shown in  FIGS. 2A through 2G . Superconducting loop  122  can include multi-crystalline d-wave superconducting material where the phase shift is caused by transition through the crystalline boundaries of the multi-crystalline material. In such embodiments, phase shifter  123  is distributed throughout portions  124  and  125  of FIG.  3 A. 
       FIG. 3B  shows a plan view of an embodiment of a N-terminal qubit  100  according to the present invention. Qubit  100  includes terminals  110 - 1  through  110 -N where superconducting loop  122  is formed between arbitrary terminals  110 -I and  110 -K. Terminals  110 -I and  110 -K can be any pair of terminals  110 - 1  through  110 -N. Qubit  100  includes phase shifter  123  in superconducting loop  122  between portion  125  and portion  124 . Multi-terminal junction  120  couples terminals  110 - 1  through  110 -N. Examples of embodiments of phase shifter  123  are shown in  FIGS. 2A through 2G . Examples of embodiments of junction  120  are shown in  FIGS. 1A through 1M . Terminals  110 - 1  through  110 -N can be made of any superconducting material compatible with the particular embodiment of phase shifter  123  and N-terminal junction  120 . Terminals  110 - 1  through  110 -N carrying superconducting currents I 1  through I N , respectively. The transport current I T  is the current directed to terminal  110 -I, I I . 
     Widths W 1  through W N , widths W J1  through W JN , and widths W S0  through W S6  are only constrained by the requirement of compatibility with phase shifter  123  and N-terminal junction  124  and, in some embodiments, typically all less than about 10 microns. Terminals  110 - 1  through  110 -N of qubit  100  as shown in  FIG. 3B  include a narrowing of the width of the terminal as the terminal is couple to junction  120 . Additionally, portions  124  and  125  are narrowed to coupled with terminals  210  and  211  of phase shifter  123 . Lengths L 1  through L N , D J1  through D JN , L S0 , and L S1  are only constrained by the requirement of compatibility with phase shifter  123  and N-terminal junction  120  and, in some embodiments, are typically all less than about 10 microns. The superconducting loop can be threaded by a magnetic flux Φ, which may contain contributions from a spontaneous supercurrent in the loop and externally applied magnetic fields. 
     N-terminal junction  120  couples one end of each of terminals  110 - 1  through  110 -N by, for example, constriction junctions, tunnel junctions, two-dimensional electron gas structures, or combinations of these. The choice of the physical sizes of the elements in N-terminal junction  120  also affects the function of qubit  100 . To achieve a small magnetic flux Φ in superconducting loop  122  (which is desirable for coherence consideration) and maximum influence of the transport currents I 1  through I N  on the properties of superconducting loop  122 , in some embodiments the terminals in the transport terminals (i.e., terminals  110 - 1  through  110 -N that are not terminals  110 -I and  110 -K) are much wider than terminals  110 -I and  110 -K at junction  120 , that is, W J1  through W JN , excluding W JI  and W JK  are much larger than W JI  and W JK . A small magnetic flux Φ can exist, even in the absence of external magnetic fields, because of spontaneous supercurrents. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system will be affected most pronouncedly by the applied transport currents I 1  through I N . 
     N-terminal qubit  100  with intrinsic phase shifter  123  can provide a basic block for construction of a qubit but can also be utilized for demonstration of macroscopic quantum tunneling and incoherent quantum noise in a solid state system. As described further below, the macroscopic quantum tunneling in a set of independent N-terminal qubits with intrinsic phase shifters (i.e., with no entanglements between individual qubits) permits construction of a random number generator that generates random series with zero correlation between numbers in the random series. 
     N-terminal qubit  100  with intrinsic phase shifter  123  includes a superconducting loop  122  linking two of the N terminals, terminals  110 -I and  110 -K. N-terminal qubit  100  with intrinsic phase shifter  123  is made superconducting by reducing the temperature of qubit  100  below the superconducting critical temperature T c  of all of the superconducting materials in qubit  100 . N-terminal junction  120  may be either symmetric or asymmetric. The superconducting materials from which qubit  100 A is constructed are constrained only by the requirement of compatibility with phase shifter  123  and N-terminal junction  120  and otherwise may have any pairing symmetry. For example, materials used may be s-wave, for example, niobium or aluminum, or d-wave, such as a high-Tc cuprate, for example YBa 2 Cu 3 O 7-x , or any superconducting material in which the Cooper pairs are in a state with non-zero orbital angular momentum. 
     N-terminal qubit  100  with intrinsic phase shifter  123  can be formed on an insulating substrate such as, for example, strontium titanate or sapphire. Phase shifter  123  may be any structure that shifts the phase of the superconducting order parameter in transition across the structure. Examples of embodiments of phase shifter  123  are shown in  FIGS. 1A through 1G . Additionally, phase shifter  123  can be incorporated into portions  124  and  125  if portions  124  and  125  are of a multi-crystalline d-wave superconducting material, where the phase shift is caused by transition through the crystalline boundaries of the multi-crystalline material. 
       FIG. 3C  shows a schematic diagram of qubit  100 . Terminals  110 - 1  through  110 -N are coupled at one end through junctions  140 - 1  through  140 -(N−1) in multi-terminal junction  120 . Terminals  110 -I and  110 -K are coupled to portions  124  and  125 . Phase shifter  123  is included in superconducting loop  122 . 
       FIG. 4A  shows a plan view of an embodiment of a qubit array  400 . Qubit array  400  includes a series of multi-terminal qubits  401 - 1  through  401 -M, each of which includes an intrinsic phase shifter  402 - 1  through  402 -N coupled in series to terminals  410  and  411 , which are themselves coupled to form a loop. As such, qubit array  400  as shown in  FIG. 4A  is an example of a qubit register structure. Multi-terminal qubits  400 - 1  through  400 -N can be embodiments of qubit  100  as shown in  FIGS. 3A through 3C . Each of qubits  401 - 1  through  401 - 2  can be threaded by a magnetic flux Φ 480-1  through Φ 480-N , respectively. The loop formed by terminals  410  and  422  can be threaded by magnetic flux Φ 481 . Terminals  410  and  411  can be made of any superconducting materials that can be coupled to the terminals of qubits  401 - 1  through  401 -N. 
     Terminal  410  is coupled to one of the terminals of qubit  401 - 1 . Another of the terminals of qubit  401 - 1  is coupled to a terminal of qubit  401 - 2 . Each of qubits  401 - 2  through  401 -N are coupled to a terminal of qubit  401 - 1  through  401 -(N−1). Additionally, a second terminal of qubit  401 -N is coupled to terminal  411 . Connectors joining qubits  401 - 1  through  401 -N can be made of any superconducting material compatible with qubits  401 - 1  through  401 -N. Superconducting current I L  can exist in the loop formed by joining terminals  410  and  411 . Widths W L1  through W L(N+1)  and W SL1  through W SL3  are constrained only by the requirement of compatibility with qubits  400 - 1  through  400 -N and are typically all less than about 10 microns. Lengths L SL1  through L SL3  and L L1  through L L(N+1)  are not critical but, in some embodiments, are typically all less than about 10 microns. 
     The magnetic flux Φ 481 , which threads the loop formed by joining terminals  410  and  411 , may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. The magnetic fluxes Φ 480-1  through Φ 480-N , which thread the loops in multi-terminal qubits  401 - 1  through  401 -N, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. 
     In accordance with embodiments of aspects of the present invention, qubit array  400  as shown in  FIG. 4A  demonstrates coupling of different qubit quantum systems (i.e., qubits  401 - 1  through  401 -N) and reading out information about this coupling. In embodiments where there is no externally applied magnetic field, the magnetic flux Φ 481  threading the loop formed by joining terminals  410  and  411  is a known function of the quantum states of the qubits  401 - 1  through  401 -N. A discussion of the relationship between the magnetic flux Φ 481  and the quantum states of qubits  401 - 1  through  401 -N is given in Appendix A, herein incorporated by reference in its entirety. 
     Therefore, measurement of the flux Φ 481 , for example by an external measuring instrument such as a magnetic force microscope, scanning SQUID microscope or scanning Hall probe, provides information on the quantum states of qubits  401 - 1  through  401 -N. In addition, application of time-dependent external magnetic fields to the superconducting loops of SQUID qubits  401 - 1  through  401 -N and to the loop formed by joining terminals  410  and  411  can function as an operating system to perform specific algorithms using the qubit register architecture of qubit array  400  as disclosed in FIG.  4 A. The relation of the time-dependent magnetic field application and the algorithm performed is described in Appendix A. 
       FIG. 4B  shows a plan view of an embodiment of a qubit array  400 . Qubit array  400  includes series coupled qubits  401 - 1  through  401 -N coupled to terminals  410  and  411 . Each of qubits  401 - 1  through  401 -N includes a superconducting loop  122  with intrinsic phase shifter  402 - 1  through  402 -N, respectively. Terminals  410  and  411  are coupled to an external source of transport current to provide current I T . Multi-terminal qubits  401 - 1  through  401 -N are embodiments of aspects of the invention described in  FIGS. 3A through 3C  and can be any multi-terminal qubit. Qubits  401 - 1  through  401 -N can be threaded by magnetic fluxes Φ 480-1  through Φ 480-N , respectively. 
     Terminals  410  and  411  can be made of any superconducting material compatible with the choice of qubits  401 - 1  and  401 -N. Connectors joining qubits  401 - 1  through  401 -N can be made of any superconducting material compatible with the choice of qubits  401 - 1  through  401 -N. Superconducting current I T , which can be present in terminals  410  and  411 , typically arises from an external transport current source. 
     Widths W L1  through W L(N+1)  and W SL1  through W SL3  are only constrained by the requirement of compatibility with qubits  401 - 1  through  401 -N and, in some embodiments, are typically all less than about 10 microns. Lengths L L1  through L L(N+1) , L SL1  and L SL3  are not critical but are typically all less than about 10 microns. The magnetic fluxes Φ 480-1  through Φ 480-N  which thread the superconducting loops in qubits  401 - 1  through  401 -N, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. 
     In accordance with an embodiment of an aspect of the invention, qubit array  400  of  FIG. 4B  demonstrates an apparatus for coupling different qubit quantum systems and reading out information about this coupling. In the case where there is no externally applied transport current (i.e., no transport current through other ones of the terminals of qubits  401 - 1  through  401 - 2  other than those coupled to terminals  410  and  411 ), the current I T  in terminals  410  and  411  is a known function of the quantum states of qubits  401 - 1  through  401 -N, as discussed in Appendix A. Therefore measurement of the current I T , for example by an external measuring instrument such as a single electron transistor, provides information on the logical states of qubits  401 - 1  through  401 -N. In addition, application of time-dependent externally applied transport current I T (t) to terminals  410  and  411  and time-dependent external magnetic fields B 1 (t) through B N (t) to the superconducting loops of qubits  401 - 1  through  401 -N, respectively, can function as an operating system to perform specific algorithms using the qubit register architecture of qubit array  400  shown in FIG.  4 B. The relation of the time-dependent magnetic field B(t) and transport current I T (t) application and the algorithm performed is described in Appendix A. 
       FIG. 5  shows a plan view of an qubits  500  and  501  coupled at junction  120  where junction  120  is, in this example, a six-terminal junction. Qubits  500  and  501 , then, share one junction  120 . Qubit  500  includes a superconducting loop coupled into junction  120  with terminals  110 - 5  and  110 - 6 . Qubit  501  includes a superconducting loop coupled into junction  120  with terminals  110 - 2  and  110 - 3 . Terminals  110 - 1  and  110 - 4  can be coupled to a source of transport current I T . Further, qubits  500  and  501  include phase shifters  523 - 1  and  523 - 2 , respectively. 
     Examples of embodiments of phase shifters  523 - 1  and  523 - 2  are shown as phase shifter  123  in  FIGS. 2A through 2G . Examples of embodiments of junction  120  are shown in  FIGS. 1A through 1M . Examples of embodiments of qubits  500  and  501  are shown in  FIGS. 3   a  through  3 C. 
     The superconducting loops of qubits  500  and  501  can be threaded by magnetic fluxes Φ 580  and Φ 581 , respectively. Terminals  110 - 1  through  110 - 6  can be made of any superconducting material compatible with the choice of qubits  500  and  501 . Superconducting current I T  in terminals  110 - 1  and  110 - 4  typically arises from an external transport current source. The superconducting loops of qubits  500  and  501  carry superconducting currents I Q0  and I Q1 , respectively. The physical dimensions of terminals  110 - 1  through  110 - 6  are constrained only by the requirement of compatibility with qubits  500  and  501  and, in some embodiments, are typically all less than about 10 microns. The magnetic fluxes Φ 580  and Φ 581  which thread the superconducting loops of qubits  500  and  501 , respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. 
     In accordance with an embodiment of an aspect of the invention, qubits  500  and  501  as shown in  FIG. 5  demonstrates an apparatus for coupling two different qubit quantum systems (i.e., the quantum systems of qubits  500  and  501 ) and reading out information about this coupling. In the case where there is no externally applied transport current, the current I T  in terminals  110 - 1  and  110 - 4  is a known function of the quantum states of qubits  510  through  511 , as discussed in Appendix A. Therefore measurement of the current I T , for example by an external measuring instrument such as a single electron transistor, provides information on the quantum states of qubits  500  and  501 . In addition, application of time-dependent externally applied transport current I T (t) to terminals  110 - 1  and  110 - 4  and time-dependent external magnetic fields B(t) to the superconducting loops of qubits  500  through  501  can function as an operating system to perform specific algorithms using the qubit register architecture qubits  500  and  501  as shown in FIG.  5 . The relation between the time-dependent magnetic field and transport current application and the algorithm performed is described in Appendix A. 
       FIG. 6A  shows a plan view of an embodiment of a qubit array  610 . Qubit array  690  includes qubits  600 - 1  through  600 -(N+1) coupled by six-terminal junctions  601 - 1  through  601 -N. Examples of embodiments of junctions  601 - 1  through  601 -N are shown as junction  120  in  FIGS. 1A through 1M . The superconducting loop of each of qubits  600 - 2  through  600 -N includes two of junctions  600 - 1  through  600 -N whereas the superconducting loops of qubits  600 - 1  and  600 -(N+1) includes one of junctions  600 - 1  and  600 -N. Each of Junctions  601 - 1  through  601 -N, as shown in  FIG. 6A , includes six terminals labeled 1 through 6 in counter-clockwise fashion. In an arbitrary one of junctions  601 - 1  through  601 -N, terminals  1  and  4  are coupled to receive transport currents I T1  through I TN , respectively. Terminals  2  and  3  form part of a superconducting loop (e.g., terminals  2  and  3  of junction  601 - 2  forms part of the superconducting loop of qubit  600 - 3 ). Terminals  5  and  6  form part of a separate superconducting loop (e.g., terminals  5  and  6  of junction  601 - 2  forms part of the superconducting loop of qubit  600 - 2 ). The superconducting loop of qubits  600 - 1  through  600 -(N+1) each include a phase shifter  602 - 1  through  602 -(N+1), respectively. Examples of embodiments of phase shifter  602 - 1  through  602 -(N+1) are shown as phase shifter  123  in  FIGS. 2A through 2G . 
     The superconducting loop of qubits  600 - 1  through  600 -(N+1) can be threaded by magnetic fluxes Φ 680-1  through Φ 680-(N+1) , respectively. The superconducting loops of qubits  600 - 1  through  600 -N can include any superconducting material compatible with junctions  601 - 1  through  601 -N and phase shifters  602 - 1  through  602 -(N+1). Terminals  1  and  4  of each of junctions  601 - 1  through  601 -N can carry superconducting currents I T1  through I TN , which typically arise from external transport current sources. The superconducting loops of each of qubits  600 - 1  through  600 -(N+1) can carry superconducting currents L SL-1  through L SL-(N+1) , respectively. All length and width scales are constrained only by the requirement of compatibility with junctions  601 - 1  through  601 -N and phase shifters  602 - 1  through  602 -(N+1), as has been previously discussed. The magnetic fluxes Φ 680-1  through Φ 680-(N+1)  which thread the loops in multi-terminal qubits  600  through  600 -(N+1) respectively may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. 
     In accordance with an embodiment of an aspect of the invention, qubit array  610  as shown in  FIG. 6A  demonstrates an apparatus for coupling a plurality of different qubit quantum systems and reading out information about the quantum states of the qubit quantum systems. In the case where there are no externally applied transport currents, the currents I T1  through I TN  in terminals carried by terminals  1  and  4  of junctions  601 - 1  through  601 -N are known functions of the logical states of the qubits  600 - 1  through  600 -(N+1), as described in Appendix A. Therefore measurement of the currents I T1  through I TN , for example by external measuring instruments such as single electron transistors, provides information on the logical states of the qubits  600 - 1  through  600 -(N+1). In addition, application of time-dependent externally applied transport currents I T1 (t) through I TN (t) to terminals  1  and  4  of junctions  601 - 1  through  601 -N, respectively, and time-dependent external magnetic fields to the superconducting loops of qubits  600 - 1  through  600 -(N+1) can function as an operating system to perform specific algorithms using the qubit register architecture of qubit array  610  as shown in FIG.  6 A. The relation of the time-dependent magnetic field and transport current application and the algorithm performed is described in Appendix A. Readout  605  reads the quantum states of each of qubits  600 - 1  through  600 -(N+1). 
       FIG. 6B  shows a plan view of an embodiment of a qubit register  680  from an array of M-qubit registers  690 - 1  through  690 -M. An example of an embodiment of qubit registers  690 - 1  through  690 -M is shown as qubit array  610  of FIG.  6 A. Registers  690 - 1  through  690 -M are coupled together through junctions  695 - 11  through  695 -(M−1)N. With reference to  FIG. 6A , terminals  1  and  4  of each of junctions  601 - 1  through  601 -N for a particular one of registers  690 - 1  through  690 -M are coupled to counterpart terminals  4  and  1 , respectively, of junctions  601 - 1  through  601 -N adjoining ones of registers  690 - 1  through  690 -M. In the embodiment shown in  FIG. 6B , each of junctions  695 - 1 ,  1  through  695 -(M−1), N are four-terminal junctions, examples of which are given as junction  120  in  FIGS. 1A through 1M . In  FIG. 6B , the four terminals of each of  695 - 1 ,  1  through  695 -(M−1), N are labeled counterclockwise as terminals  1 - 4 , as shown for junction  695 - 1 ,  1 . As an example, terminal  3  of junction  695 - 1 ,  1  is coupled to terminal  1  of junction  601 - 1  of array  690 - 1  and terminal  1  of junction  695 - 1 ,  1  is coupled to terminal  4  of junction  601 - 1  of array  690 - 1 . Terminal  2  is coupled to terminal  4  of junction  695 - 1 ,  2 . Terminal  4  of junctions  695 - 1 ,  1  through  695 -(M−1),  1  can be coupled to current sources to receive currents I A1  through I A(M−1) , respectively. Further, terminal  1  of each of junctions  601 - 1  through  601 -N of array  690 -M can be coupled to receive currents I B1  through I BN , respectively. All length and width scales are constrained only by the requirement of compatibility with qubit registers  690 - 1  through  690 -M as described in FIG.  6 A. 
     Magnetic fluxes Φ 680-1,1  through Φ 680-(M),(N)  can be embraced by superconducting loops contained in qubit registers  690 - 1  through  690 -M. The magnetic fluxes Φ 680-11  through Φ 680-(M)(N)B  which thread the superconducting loops in qubits  600 - 1  through  600 -N of each of qubit registers  690 - 1  through  690 -M, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. 
     In accordance with an embodiment of an aspect of the invention, the array of registers  690 - 1  through  690 -M shown in  FIG. 6B  demonstrates an apparatus for coupling a plurality of different qubit quantum systems and reading out information about the logical states of the qubit quantum systems. In the case where there are no externally applied transport currents, the currents I A1  through I A(M−1)  and currents I B1  through I B(N−1)  are known functions of the quantum states of the qubits in qubit registers  690 - 1  through  690 -M, as discussed in Appendix A. Therefore measurement of the currents I A1  through I A(M−1)  and I B1  through I B(N−1) , for example by external measuring instruments such as single electron transistors, provides information on the quantum states of the qubit  600 - 1  through  600 -N in each of qubit registers  690 - 1  through  690 -M. In addition, application of time-dependent externally applied transport currents I A1 (t) through I A(M−1) (t) to terminal  4  of junctions  695 - 11  through  695 -(M−1) 1 , respectively, and I B1 (t) through I B(N−1) (t) to terminal  1  of junctions  601 - 1  through  601 -N of qubit register  690 -M, respectively, and time-dependent external magnetic fields B(t) to the superconducting loops of qubits  600 - 1  through  600 -N of each of qubit register  690 - 1  through  690 -M function as an operating system to perform specific algorithms on qubit register  680  as shown in FIG.  6 B. The relation of the time-dependent magnetic field and transport current application and the algorithm performed is described in Appendix A. 
     In accordance with another embodiment of an aspect of the present invention,  FIG. 7  demonstrates an apparatus for measuring voltages  700 . Apparatus  700  is a radio-frequency single electron transistor electrometer and is well-known and described, for example, in A. N. Korotkov and M. A. Paalanen, “Charge Sensitivity of Radio-Frequency Single Electron Transistor,  Appl. Phys. Lett . 74, 26 (1999), which is herein incorporated by reference in its entirety. Apparatus  700  can be utilized in readout  605  of  FIG. 6A  to read the quantum state of each of qubits  600 - 1  through  600 -(N+1). 
     The single-electron transistor (SET)  709  can be made of any material that displays a Coulomb blockade effect, for example niobium, aluminum, lead, tin, and any high-temperature superconducting cuprate. P. Joyez, P. Lafarge, A. Filipe, D. Esteve, and M. H. Devoret, “Observation of Parity-Induced Suppression of Josephson Tunneling in the Superconducting Single Electron Transistor”,  Phys. Rev. Letters , Vol. 72, No. 15, 2458-61 (Apr. 11, 1994), describes operation and manufacture of single electron transistors and is incorporated by reference herein in its entirety. SET  709  is placed in a high quality factor tank circuit  712  tuned to resonance. Tank circuit  712  includes inductor  707  and capacitor  708 . Capacitor  708  is coupled across SET  709 . A third terminal of SET  709  is coupled to electrode  710 . A radio-frequency or microwave signal  704  is introduced into the circuit  712 . The reflected signal  705  is a strong function of the voltage difference between electrode  710  and ground  711 . Analysis of reflected signal  705  using established techniques allows measurement of the voltage difference between electrode  710  and ground  711 . 
     Read-out of the state of the qubit quantum system may be done via the use of a single electron transistor (SET)  709  according to known procedures, for example, as described in R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing, and D. E. Prober, “The Radio-Frequency Single-Electron Transistor (RF-SET): A Fast and Ultrasensitive Electrometer,”  Science , Vol. 280, 1238-42 (May 22, 1998). SET  709  may be coupled to three devices (e.g., terminals  710 ,  711  and  712 ). An electron or Cooper pair can tunnel onto SET  709  when SET  709  is uncharged. However, SET  709  is small enough that once an electron or Cooper pair tunnels onto SET  709 , the charging of SET  709  electrically repels and prevents further tunneling onto SET  709 . A terminal  710  associated with SET  709  can change the voltage of SET  709  and de-tune tank circuit  712 , changing the characteristics of the reflected wave  705 . 
     In operation, in order to measure a current, for example one of currents I A1  through I A(M−1)  or I B1  through I B(N−1)  shown in  FIG. 6B , electrode  710  is coupled to terminal  4  of junctions  495 - 11  through  495 -(M−1)N or terminal  1  of junctions  601 - 1  through  601 -N of qubit register  690 -M. The current at those terminals can be measured by applying signal  704  and monitoring signal  705  of FIG.  7 . 
     All structures in  FIGS. 1A through 7  can be formed using conventional fabrication techniques. Elemental s-wave superconductors may be purchased from, for example, CIL Cambridge Isotopes Laboratories Inc. and substrates upon which superconducting structures can be formed may be purchased, for example, from Kagaku Gijutsu-sha of Tokyo, Japan or from Shinkosha, Ltd. (c/o Nikko Trading Co.). The fabrication process for a four-terminal structure with s-wave superconductors is developed in detail, for example, in B. J. Vleeming, “The Four-Terminal SQUID,” Ph.D. Dissertation, Leiden University, The Netherlands, 1998 and references therein, which has previously been incorporated into this disclosure by reference. 
     Some embodiments of the invention allow all of the operations that are required for quantum computing to be done without the application of external magnetic fields. Operations such as read and initialization, as well as operating system gates such as application of Pauli operators σ x  and σ z , and furthermore operations for maintaining coherence in the state of the qubit can be achieved, without the use of external magnetic fields. 
     In some embodiments, as illustrated in  FIG. 8 , a qubit  100  comprises a five terminal junction  120  and a superconducting loop  122  with phase shifter  123 . The embodiment of qubit  100  shown in  FIG. 8  includes terminals  110 - 1  through  110 - 5 , where terminals  110 - 4  and  110 - 5  are connected to form superconducting loop  122  by a phase shifter  123 . Terminals  110 - 1  through  110 - 3  can be utilized for the application of transport current as well as for voltage measurements in reading the quantum state of qubit  100 . Five terminal junction  120  can be any junction for coupling terminals  110 - 1  through  110 - 5 , such as those discussed with  FIGS. 1A through 1M . For example, junction  120  can be a two dimensional electron gas junction and terminals  110 - 1  through  110 - 5  can be of niobium. A controller  800  can be coupled with terminals  110 - 1  through  110 - 3  in order to supply currents to and measure voltage across various ones of terminals  110 - 1  through  110 - 3 , as discussed below. Qubit  100  as shown in  FIG. 8 , then, can be both symmetric and asymmetric. 
     Although qubit  100  in  FIG. 8  is being discussed as a five-terminal qubit, one skilled in the art will recognize that methods and examples given here are extendable to qubit  100  having any number of terminals  110 - 1  through  110 -N. 
     The quantum state of the quantum system of qubit  100  of  FIG. 8 , for example, can be initialized by controller  800  passing a transport current I T  through junction  120  for a sufficient duration, as is further discussed in Appendix A. The transport current I T  through junction  120  has the effect of biasing the energy states of the quantum system of qubit  100  towards a particular energy state and of breaking the ground state degeneracy. Given a sufficient period of time, the quantum system of qubit  100  will relax into the lower energy state. By manipulating the direction of the transport current I T  passing through junction  120 , qubit  100  can be initialized such that the quantum state of qubit  100  is the desired state. For example, controller  800  can apply a bias current from terminal  110 - 1  to terminal  110 - 3 , thereby selects a particular state of the quantum system of qubit  100 . By reversing the current through the same terminals, controller  800  can select the opposite state of the quantum system. 
     A read operation on qubit  100  of  FIG. 8  can be accomplished based on the fact that each of the two degenerate ground states of the quantum system of qubit  100  exhibits a unique current-voltage curve with respect to current flowing between terminals  110 - 1  and  110 - 3 . Each of the two degenerate ground states results in a different critical current in junction  120 . The critical current is the current which, if exceeded, results in junction  120  developing a resistance. Therefore, determining which of the two critical currents is appropriate for junction  120  differentiates between the two degenerate ground states of the quantum system. 
     The quantum state of the quantum system of qubit  100  of  FIG. 8 , for example, can be read by controller  800  passing a transport current I T  through junction  120  (for example between terminals  110 - 1  and  110 - 3 ). The critical current I C  of junction  120  is dependent on the quantum state of the quantum system of qubit  100 , with one state corresponding to a lower value of critical current in junction  120  and the opposite state corresponding to a higher value of critical current. In any Josephson junction, if the critical current I C  is exceeded, dynamical effects result and a resistance becomes present in the junction. Thus, determining the state of the quantum system of qubit  100  can be accomplished by discerning the value of the critical current I C  in the junction, see Appendix A. 
     In one method of measuring the quantum state of the quantum system of qubit  100 , controller  800  applies a transport current I T  to junction  120  which is between the known upper and lower critical current values (i.e., between the values of the critical current I C  for each of the quantum states). The upper and lower values of the critical current I C  is dependent upon the particular embodiment of qubit  100 . When the transport current I T  is applied, if the system occupies the state associated with the lower critical current, then the transport current will have exceeded the critical current value of the junction, thus resulting in a junction resistance, and a corresponding voltage across the terminals (for example, between terminals  110 - 1  and  110 - 3  to which the transport current I T  is applied). Alternatively, if the system occupies the high critical current state, no voltage across the terminals will result. Controller  800 , then, can determine the quantum state of the quantum system of qubit  100  by monitoring the voltage across junction  120  while applying the transport current I T  through junction  120 . For example, by applying a transport current from terminal  110 - 1  through terminal  110 - 3  of qubit  100  of  FIG. 8 , and measuring the voltage between terminals  110 - 1  and  110 - 3 , it is possible for controller  800  to determine the state of the system, where the presence of a voltage indicates one of the states and the absence of voltage represents the opposite state. In some embodiments, controller  800  blocks the flow of current through terminal  110 - 2  by, for example, shorting terminal  110 - 2 . 
     A phase gate operation σ z  can be performed, for example, on qubit  100  of  FIG. 8  by controller  800  applying a transport current I T  pulse through junction  120 . In essence, a quantum gate operation operates to bias a particular state of the quantum system of qubit  100 . Thus, in action, the current pulse producing the σ z  operation can be similar to the read operation discussed above, except that the magnitude of the current pulse applied is much less than the magnitude used for the exemplary read operation so that no dynamical effect results from its application, see Appendix A. The magnitude and duration of the transport current I T  pulse necessary to affect the σ z  operation is specific to the particular embodiment of qubit  100 . As long as the magnitude of the pulse of transport current I T  is small, application of the transport current I T  by controller  800  will not destroy the quantum superposition of states in the quantum system of qubit  100 , but merely weight one of the states as desired. 
     A phase gate operation σ x  can be performed, for example, on qubit  100  of  FIG. 8  by application of a transport current I T  to terminal  110 - 2  and allowing it to escape through both of terminals  110 - 1  and  110 - 3 . In performing the σ x  operation, then, controller  800  applies a transport current to junction  120  through terminal  110 - 2 , and having current flow out of junction  120  through terminals  110 - 1  and  110 - 3 . Application of the transport current in this manner results in manipulating the height of the potential barrier that separates the two degenerate states of the quantum system of qubit  100 , see Appendix A. The magnitude of the transport current applied is dependent on the actual configuration of qubit  100 . When applied for a short duration of time, the height of the potential barrier between the two degenerate states is reduced and the tunneling frequency of the system increases for the duration of the pulse. 
     Furthermore, it is possible to tune the tunneling frequency of the qubit by applying a steady state current in the same manner as that of the σ x  operation. This allows a tuning of the quantum overlap of the two degenerate ground states in the qubit to a desired range. This is useful in an array of qubits where the tunneling frequencies of some or all of the qubits vary. By tuning the tunneling frequency of each qubit, the array can be tuned into a uniform range of tunneling frequencies, thus allowing more predictable application of quantum algorithms in the array. 
     Tuning can be achieved for example on qubit  100  of  FIG. 8  by application of a steady state current to terminal  110 - 2  and allowing the current to escape through terminals  110 - 1  and  110 - 3 . A method for tuning the tunneling frequency of a qubit would comprise controller  800  applying a steady state current at terminal  110 - 2  of junction  120 , while grounding or allowing escape from the adjacent terminals  110 - 1  and  110 - 3 . By varying the magnitude of the applied steady state current, the tunneling frequency of the qubit can be manipulated. 
       FIG. 9  shows a qubit array  900  including qubits  100 - 1  and  100 - 2 . Array  900  can include any number of qubits. Qubits  100 - 1  and  100 - 2 , for exemplary purposes only, are each five-terminal qubits  100  as shown in FIG.  8 . Qubits  100 - 1  and  100 - 2  are entangled by junction  145  coupled between superconducting loop  122 - 1  of qubit  100 - 1  and superconducting loop  122 - 2  of qubit  100 - 2 . In order that qubits  100 - 1  and  100 - 2  can behave independently, it is desirable that junction  145  can be opened to isolate qubit  100 - 1  from qubit  100 - 2  (and hence from all the other qubits in qubit array  900 ) and closed to entangle qubit  100 - 1  with  100 - 2 , see Appendix A. Junction  145  can be controlled by controller  800  by applying a voltage to junction  145 , for example capacitively through plate  146 . 
       FIGS. 10   a  and  10   b  show operation of plate  146  in opening and closing junction  145  and thereby entangling or isolating qubits  100 - 1  and  100 - 2  of array  900 . Controller  800  is electrically coupled to plate  146  at a terminal so that controller  800  can apply a voltage to plate  146 , which is capacitively coupled to junction  145 . Junction  145  can be any superconducting junction, including a two-dimensional electron gas, tunneling junction, or constriction junctions. 
     As shown in  FIG. 10   a , when controller  800  does not apply a voltage to plate  146 , current can freely flow between superconducting loop  122 - 1  of qubit  100 - 1  and superconducting loop  122 - 2  of qubit  100 - 2 . Therefore, qubits  100 - 1  and  100 - 2  are entangled. 
     As shown in  FIG. 10   b , when controller  800  applies a voltage to plate  146 , electrons are prevented from flowing through junction  145  by electric fields  143  and junction  145  effectively becomes opened, thus isolating qubit  100 - 1  from qubit  100 - 2 . 
       FIG. 11  illustrates a qubit array  900  of five terminal qubits, where each of qubits  100 - 1  through  100 -N is coupled by a junction  145 - 1  through  145 -(N−1) to its nearest neighbors. That is, qubit  100 - 1  through  100 -N can be entangled by junction  145 - 1  and the entanglement can be switched by plate  146 - 1 . Further, qubit  100 -(N−1) and qubit  100 -N can be entangled by junction  145 -(N−1) and the entanglement can be switched by plate  146 -(N−1. 
     Although array  900  of  FIGS. 9 and 11  illustrate entanglements between qubits in a linear array, one skilled in the art should recognize that a two-dimensional array of qubits  100  can be entangled in this fashion. Each superconducting loop  122  can be switchably entangled to any number of other superconducting loops through a junction  145  with a plate  146 . 
       FIG. 12  shows a particular embodiment of qubit array with qubits  100 - 1  through  100 -N coupled by junctions  145 - 1  through  145 -(N−1) and switched with plates  146 - 1  through  146 -(N−1), respectively. In  FIG. 12 , junctions  145 - 1  through  145 -(N−1) are each two-dimensional electron gas junctions. Additionally, junction  145 - 1 , for example, is coupled such that it is within both superconducting loop  121 - 1  and  121 - 2 , rather than simply linking superconducting loops  121 - 1  through  121 - 2  as is shown in FIG.  11 . In this fashion, junction  145 - 1 , through plate  146 - 1 , controls the coupling of superconducting loop  121 - 1  with multiterminal junction  120 - 1 , controls the coupling of superconducting loop  121 - 2  with multiterminal junction  120 - 2 , and controls the entanglement between qubits  100 - 1  and  100 - 2 . To avoid any interaction of the entangling junction with the flux in the qubit, the entangling junctions  145 - 1  through  145 -(N−1) are isolated from the main loop of the qubit by a distance W 145 . This further decouples the qubit from any interaction with the environment or surrounding fields, thus decreasing the decoherence in the qubit. 
     In operation, qubits and qubit arrays according to the present invention (such as, for example, the embodiments discussed above) are cooled to a temperature well below the superconducting transition temperature T c  of all superconducting materials utilized to fabricate the particular structures. In an exemplary embodiment, the structures described in  FIGS. 1A through 13  are cooled to an operating temperature of about 10 milliKelvin so that all structures are superconducting, all phase shifters are operative and decoherence processes due to thermal fluctuations and inelastic scattering are suppressed. 
     In accordance with current theoretical descriptions, for example, the Eliashberg theory of superconductivity (see, e.g., R. de Bruyn Ouboter and A. N. Omelyanchouk, “Macroscopic Quantum Interference Effects in Superconducting Multiterminal Structures,”  Superlattices and Microstructures , Vol. 25, No. 5/6 (1999)) an order parameter Ψ describes current flow in superconductors and phase differences in multi-terminal junctions. Multi-terminal qubit  100  with intrinsic phase shifter  123  as shown, for example, in  FIGS. 3A through 3C  have degenerate ground states of the qubit quantum system, the supercurrent circulating in superconducting loop  122  of qubit  100  being twice degenerate if no external magnetic field or transport current from any external current source is applied. The two degenerate states having the ground state energy and definite magnetic moment, |0&gt; and |1&gt;, correspond to minimal supercurrents circulating through superconducting loop  122  in clockwise and counter-clockwise senses. The two states associated with the supercurrent in superconducting loop  122  permit quantum computing in the standard fashion, which is described in many papers and books (see, e.g., J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, and Seth Lloyd, “Josephson Persistent-Current Qubit,”  Science  285, 1036-39 (1999)). 
     The role of phase shifter  123  in multi-terminal qubit  100  is to cause the two basis states of the qubit to be naturally degenerate. This is a major advantage over other qubit designs, for example that of Mooij et al.,  Science  285 1036, where it is necessary to apply a magnetic field in order to bring the basis states of the qubit quantum system into resonance (i.e., cause them to be degenerate). The magnetic field required by the system of Mooij et al.,  Science  285 1036, needs to be extremely finely tuned in order to maintain the resonance condition. This is because the precision to which the external field has to be tuned is approximately the tunneling amplitude Δ T  between qubit basis states which is usually about 5 GHz and corresponds to a magnetic field precision of one part in about 10 6 . 
     One application of embodiments of multi-terminal qubits according to the present invention is a quantum computational random number generator. As a random number generator, the quantum states of an array of qubit  610  as shown in  FIG. 6A  or array  680  as shown in  FIG. 6B  or of array  900  as shown in  FIGS. 9 and 11  evolve to a state where the qubit quantum system on each of individual qubits (e.g, qubits  600 - 1  through  600 -(N+1) of array  610 ) has an equal (or at least known) probability of evolving to each of the basis states |0&gt; and |1&gt;. The basis states, which are related to the superconducting current directions in the superconducting loops, are then determined, for example, by observing each of qubits with a magnetic force microscope, a SQUID magnetometer, a scanning Hall probe, or other magnetic probes, or alternately by applying transport currents and measuring voltage drops with, for example, apparatus  700  as described in  FIG. 7 , which will fluctuate if the state to be measured is in the higher energy state or remain static otherwise. Each determined state (clockwise current or counterclockwise current) corresponds to a bit value (0 or 1) so that the collection of determined states provides a random binary value having as many bits as there are qubits in the array. Quantum theory indicates that, with known (including zero) entanglements between individual qubits, a series of bits thus generated can be random without correlation or repetition between bits. 
     Qubits according to embodiments of aspects of the current invention may also alternatively be read by other readout devices such as a magnetic force microscope (MFM) tip, a superconducting quantum interferometer device (SQUID) loop, or a Hall probe device. The readout device measures the weak local magnetic fields that the spontaneous supercurrents (clockwise or counterclockwise) cause in the vicinity of the multi-terminal qubit. More particularly, the MFM scans a microscopic magnetized tip attached to a cantilever across the surface and measures deformation of the cantilever as the mechanical force that acts on the magnetized tip. Alternatively, a superconducting loop can detect the magnetic flux in the vicinity of the multi-terminal qubit. Alternatively, a Hall probe can detect the magnetic flux in the vicinity of the multi-terminal qubit. Another possible read out system may use a difference in the absorption of circularly polarized microwave radiation due to the clockwise or counterclockwise currents by a multi-terminal qubit. 
     The time required for a calculation and the interpretation of the read out results depends on the calculation performed. Such issues are the subject of many papers on quantum computing, for example P. Shor, “Introduction to Quantum Algorithms,” Los Alamos preprint server condmat/005003 (Apr. 29, 2000). The structures described herein can serve as components of quantum computing systems and also can implement any single qubit algorithm. 
     In general a controller can be electrically coupled to provide current to the terminals of each qubit  100  in an array of qubits (for example, controller  800  shown in  FIGS. 8 ,  9  and  11 ). Controller  800  controls the currents in each of the terminals and thereby is capable of controlling the initial states of each qubit, the entanglements between qubits, the application of magnetic fields to each qubit, and the measurements of the state of the qubit. As such, one skilled in the art will recognize that controller  800  can be a microprocessor based system operating software which controls the programming and readout of qubits single or in arrays as shown in  FIGS. 6A ,  6 B,  6 C,  9 , and  11 . 
     Although the invention has been described with reference to particular embodiments, the description is exemplary only and should not be considered limiting. One skilled in the art may recognize several obvious variations, which are intended to be within the scope and spirit of the present disclosure. One skilled in the art will recognize embodiments of other qubits according to the present invention which are within the scope of this disclosure. Various adaptations and combinations of features of the embodiments disclosed are within the scope of the invention. As such, the invention is limited only by the following claims.