Abstract:
We propose a practical method to generate cluster states for quantum computers. The qubit systems can be NV-centers in diamond, Pauli-blockade quantum dots with an excess electron or ion traps with optical transitions, which are subsequently entangled using a so-called double-heralded single-photon detection scheme. The fidelity of the resulting entanglement is extremely robust against the most important practical errors such as detector loss, light collection efficiency and mode mismatching. The cluster states are generated efficiently using a modified probabilistic teleportation protocol.

Description:
BACKGROUND  
       [0001]     Many physical implementations of quantum computers have been proposed. Knill, Laflamme, and Milburn, Nature 409, 26 (2001), for example, proposed a quantum computing system using optical qubits manipulated with linear optics. Alternatively, optical quantum information systems can use the “cluster state model” of quantum computation introduced by R. Raussendorf and H. Briegel, Phys. Rev. Lett. 86, 5188 (2001). The cluster state model requires the generation of a specific entangled state (or the cluster state), augmented by single qubit measurements in arbitrary bases, or, equivalently, arbitrary single qubit operations together with single-qubit measurements in a particular basis. Linear optical quantum computing (LOQC) systems can eliminate the need for a direct interaction between photonic qubits by using measurement-induced nonlinearities, but generally require a quantum memory for photons, which may be difficult to implement.  
         [0002]     Some other particularly promising proposals for quantum computing systems implement unitary operations and readout in matter qubits via laser-driven optical transitions. Examples of these systems include ion-trap systems such as described by J. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995), system using nitrogen-vacancy (N-V) defects in diamond as described by Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004), and systems using the Pauli-blockade effect in quantum dots with a single excess electron as described by Pazy et al., Europhys. Lett 62, 175 (2003) or by Nazir et al., Phys. Rev. Lett. 93, 150502 (2004). However, many of the systems using optical addressing or readout of matter qubits also use a relatively short-range interaction (e.g., the Coulomb interaction) in two-qubit gates. The short effective range of the interactions requires that the interacting matter qubits be close together. In contrast, optical addressing of the qubits requires that the separations between matter qubits be large enough to that the optical pulses can resolve the qubits. The short-range interaction for 2-qubit operations and the optical addressing of the qubits therefore seem to lead to contradictory system requirements. (Quantum computing systems implemented using ion traps can avoid this contradiction but must use mini-trap arrays or moving qubits to obtain scalability.)  
         [0003]     Systems and methods for efficiently producing remote interaction of stationary qubits, for example, for high-fidelity entanglement of static qubits are thus sought.  
       SUMMARY  
       [0004]     In accordance with an aspect of the invention, a process can use photon measurements to entangle the states of two separated systems. The process generally includes exciting first and second systems using excitations capable of transforming each system from a first state of the system to an excited state of the system. A first measurement of photonic output modes corresponding to transitions from the excited states to the respective first states is performed in a manner that is unable to distinguish whether photons originated from the first or second system. The process then performs the operations capable of transforming the first states of the systems to respective second states of the systems and simultaneously transforming the second states of the systems to the respective first states of the systems. After performing these operations, the process again excites the first and second systems using the excitations that transform the first states to the excited states, before the process performs a second measurement of the photonic output modes. The second measurement is also unable to distinguish whether photons originated from the first or second system. Measurement results of the first and second measurements indicate whether the process successfully entangled quantum states of the first and second systems. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0005]      FIG. 1  shows a quantum coherent system in accordance with an embodiment of the invention using measurements of emitted photons to entangle the quantum states of two separated matter systems.  
         [0006]     FIGS.  2  is a flow diagram of an entanglement process in accordance with an embodiment of the invention using a double heralding and detectors that may be unable to reliably distinguish a single emitted photon from two emitted photons.  
         [0007]      FIG. 3  shows a quantum coherent system in accordance with an embodiment of the invention using non-absorbing parity measurements of photonic modes for entanglement of the states of matter systems.  
         [0008]      FIG. 4  is a flow diagram of an entanglement process in accordance with an embodiment of the invention using non-absorbing parity measurements.  
         [0009]      FIG. 5  shows a quantum coherent system in accordance with an embodiment of the invention using measurements of emitted photons to entangle and/or produce cluster states of multiple separated matter qubits.  
         [0010]      FIG. 6  is a flow diagram of process in accordance with an embodiment of the invention capable of producing cluster states of spatially separated matter systems.  
         [0011]      FIG. 7  illustrates a process in accordance with an embodiment of the invention that uses a multiply-heralded entanglement operation to join cluster states and form a larger cluster state. 
     
    
       [0012]     Use of the same reference symbols in different figures indicates similar or identical items.  
       DETAILED DESCRIPTION  
       [0013]     In accordance with an aspect of the invention, a distributed system uses matter systems for quantum information storage and uses electromagnetic transitions of the matter systems with photon detection to generate entangled states or cluster states of spatially separated matter systems. In accordance with further aspect of the invention, a non-deterministic entanglement operation with any success probability can entangle the states of two spatially separated matter systems or efficiently generate cluster states of many spatially separated matter systems.  
         [0014]      FIG. 1  schematically illustrates an exemplary embodiment of a quantum information system  100  in accordance with an embodiment of the invention. System  100  includes two matter systems  110 - 1  and  110 - 2 , sometimes referred to generically as matter systems  110 . Each matter system  110  has quantum states |⇑&gt; and |⇓&gt;, which may be identified with the basis or logical states |0&gt; and |1&gt; of a qubit. In an exemplary embodiment, states |⇑&gt; and |⇓&gt; are long-lived, low-lying spin projection states of a matter system such as an atom or molecule. Each matter system  110 - 1  and  110 - 2  also has an excited state |e&gt; that is coupled to only one of the states |⇑&gt; and |⇓&gt; through emission/absorption of a photon. Without loss of generality, the following assumes that excited state |e&gt; couples to state |⇓&gt;, and that the transition between excited state |e&gt; and state |⇑&gt; is forbidden, e.g., by a selection rule. Some known systems that may have suitable quantum energy levels for matter systems  110 - 1  and  110 - 2  include N-V defects in diamond such as described by F. Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004), quantum dots with a single excess electron such as described by E. Pazy et al., Europhys. Lett. 62, 175 (2003) or A. Nazir et al., quant-ph/0403225 (2004), and various trapped ion and atomic systems.  
         [0015]     Matter systems  110 - 1  and  110 - 2  are in respective optical cavities  120 - 1  and  120 - 2  sometime generically referred to herein as optical cavities  120 . Each optical cavity  120  preferably has a resonant mode corresponding to the wavelength of a photon emitted when the excited state |e&gt; decays. Optical cavities  120 - 1  and  120 - 2  are also constructed to preferentially leak electromagnetic radiation to respective photonic modes  130 - 1  and  130 - 2 , which enter an optical measurement system  150 . Leaky optical cavities  120  with a highly preferred direction of light emissions can be constructed using known quantum optics techniques.  
         [0016]     Measurement system  150  is ideally capable of measuring a single photon emitted from cavities  120 - 1  or  120 - 2  but without distinguishing whether matter system  110 - 1  or  110 - 2  is the source of the measured photon. In the illustrated embodiment, measurement system  150  includes a 50-50 beam splitter  152  having photonic modes  130 - 1  and  130 - 2  from resonant cavities  120 - 1  and  120 - 2  as input modes. Detectors  154  and  156  are positioned to receive and measure the radiation in output modes  135 - 1  and  135 - 2  of beam splitter  152 .  
         [0017]     An excitation system  160  in  FIG. 1  operates on matter systems  110 - 1  and  110 - 2  as needed to selectively drive one or both of matter systems  110 - 1  and  110 - 2  to other energy states and in particular to drive state |⇓&gt; to excited state |e&gt;. As noted above, matter systems  110 - 1  and  110 - 2  are embedded in separate optical cavities  120 - 1  and  120 - 2 , and cavities  120 - 1  and  120 - 2  can be designed such that only the transition between state |⇓&gt; and |e&gt; is coupled to the resonant mode of cavities  120 - 1  and  120 - 2 . Accordingly, excitation system  160  can produce a π-pulse that induces a transition to excited state |e&gt; from a component state |⇓&gt; but does not cause the transition to excited state |e&gt; from a component state |⇑&gt;, e.g., because of a selection rule based on a conservation law such as conservation of angular momentum or conservation of energy. Excitation system  160  may also be used to induce transitions of matter system  110 - 1  or  110 - 2  between states |⇑&gt; and |⇓&gt;, e.g., for single-qubit operations, but excitation system  160  may require separate subsystems for the different transitions because of the difference between the energy associated with the transition state |⇓&gt; to state |⇑&gt; and the energy associated with the transition of state |⇓&gt; to state|e&gt;.  
         [0018]     For illustrative purposes, the following description emphasizes an exemplary embodiment of system  100 , which uses N-V defects in diamond for matter systems  110 - 1  and  110 - 2 . An N-V defect corresponds a nitrogen (N) atom in place of a carbon (C) atom and next to a vacancy in a diamond crystal. Such defects are know have a triplet ground state corresponding to the spin projections −1, 0, and 1 of a spin-one system, and the ground states have a strongly dipole allowed optical transition to a first excited triplet state. An external magnetic field can be applied to a N-V defect to break the degeneracy of the ground state, and two of the triplet states (e.g., the ±1 spin projection states) can then be used as states |⇑&gt; and |⇓&gt;. (The third N-V defect state (e.g., with spin projection 0) is normally not required or used when matter systems  120  represent qubits having states |⇑&gt; and |⇓&gt; as basis states.) The transition between an excited state and the ground state of an N-V defect produces a photon having an optical wavelength, permitting uses of conventional optical elements and photodiodes in optical measurement system  150 .  
         [0019]     The N-V defects in an exemplary embodiment of the invention are in nanocrystallites of high-pressure, high-temperature diamond (type 1b). The nanocrystallites, which may be about 20 nm in diameter, can be embedded in resonant cavities  120 - 1  and  120 - 2 , which may be formed in and on a wafer or die, for example, as defects within a photonic crystal. The entire structure can then be kept at a temperature that provides a desired coherence time for the states of matter systems  110 . In general, the spin states of N-V defects are known to remain coherent for milliseconds at room temperature up to seconds at low temperatures (e.g., about 2° K). Excitation system  160  for N-V defects  110 - 1  and  110 - 2  preferably includes a laser capable of producing a short pulse for a π-pulse that efficiently transforms state |⇓&gt; to excited state |e&gt;.  
         [0020]     In accordance with an aspect of the invention, system  100  can employ a double-heralded or multiply-heralded entanglement process  200  illustrated in  FIG. 2  to entangle the quantum states of matter systems  110 - 1  and  110 - 2 . Process  200  is double heralded as described below in that a combination of two specific measurement results marks successful production of an entangled state of the spatially separated matter systems  110 - 1  and  110 - 2 . The double heralding makes the entanglement process robust against measurement errors.  
         [0021]     Entanglement process  200  begins as shown in  FIG. 2  with a state preparation step  210  that sets one or both matter systems  110 - 1  and  110 - 2  into known states, e.g.,  
              +   〉     i     =       1     2       ⁢     (            ↑   〉     i     +          ↓   〉     i       )           
 
 where index i takes values 1 and 2 respectively corresponding to matter systems  110 - 1  and  110 - 2 . To illustrate a definite example of an entanglement operation, following assumes that step  210  sets matter systems  110 - 1  and  110 - 2  in respective states |+&gt; 1  and |+&gt; 2  although the entanglement operation can begin with other initial states, eg.,  
              -   〉     i     =       1     2       ⁢     (            ↑   〉     i     -          ↓   〉     i       )           
 
 or any linear combination of states |⇑&gt; i  and |⇓&gt; i . With the exemplary initial states |+&gt; 1  and |+&gt; 2  of the separate matter systems  110 - 1  and  110 - 2 , the initial state |ψ 0 &gt; of in system  100  is a product state as indicated in Equation 1, where states |0&gt; 1  and |0&gt; 2  correspond to no photons in output modes  130 - 1  or  130 - 2 .  
                      ψ   0     〉     i     =         1     2       ⁢       (            ↑   〉     1     +          ↓   〉     1       )     ⊗     1     2         ⁢       (            ↑   〉     2     +          ↓   〉     2       )     ⊗          0   〉     1       ⁢          0   〉     2       =       1   2     ⁢       (            ↑   ↑     〉     +          ↑   ↓     〉     +          ↓   ↑     〉     +          ↓   ↓     〉       )     ⊗          0   〉     1       ⁢          0   〉     2                 Equation   ⁢           ⁢   1             
 
         [0022]     The specific method for preparing the initial quantum state |ψ 0 &gt; will depend on the type of matter system  110  employed. A general process for producing the desired state might include driving the matter system into any known state and then applying local unitary transformations (if necessary) to transform the known state to the desired state. In general, a matter system  110  can be set into a known state, for example, by cooling that the matter system  110  down into a non-degenerate ground state. Alternatively, optical pumping can preferentially populate an excited state, and/or the state can be measured, for example, by detecting an optical transition. Systems and processes for implementing local unitary transformations suitable for preparation of the initial state are generally required in quantum information processing systems and are well known for a variety of quantum systems.  
         [0023]     For an N-V defect, an initial state |+&gt; can be created by, for example, by cooling the N-V defect in a magnetic field to produce the lowest energy state |⇑&gt; or |⇓&gt; and then performing a single qubit operation to transform the matter system to the desired initial state, e.g., state |+&gt;. For example, application of an ESR (Electron Spin Resonance) pulse, i.e., an oscillating magnetic field pulse having a frequency that is resonant with the energy gap between states |⇑&gt; and |⇓&gt; and a pulse envelope with a length such that the N-V defect undergoes ¼ of a Rabi cycle, can transform state |⇓&gt; or |⇑&gt; to the desired state |+&gt;.  
         [0024]     Once system  100  is in the desired initial state, step  220  applies optical π-pulses to matter systems  110 - 1  and  110 - 2  to coherently pump the population of states |⇓&gt; i  in the initial state |ψ 0 &gt; into respective excited states |e&gt; i . As known in the art, a π-pulse generally is a pulse of laser light having a frequency that is resonant to the transition of state |⇓&gt; to excited state |e&gt; and a duration that induces population inversion. The excitation step  220  thus transforms initial product state |ψ 0 &gt; to a state |ψ 1 &gt; as indicated in Equation 2.  
                      ψ   0     〉     ⁢     ⟶     π   -   pulses       ⁢          ψ   1     〉       =       1   2     ⁢       (            ↑   ↑     〉     +          ↑   e     〉     +          e   ↑     〉     +        ee   〉       )     ⊗          0   〉     1       ⁢          0   〉     1               Equation   ⁢           ⁢   2             
 
         [0025]     Each excited state or states |e&gt; will decay or fluoresce by emitting a photon when transitioning to state |⇓&gt;, causing the resonant cavity  120  to later release a photon. In general, the time at which a photon is released is random time but statistically characterized by of the half-lives of excited state |e&gt; and the resonant cavity  120 . Step  230  of process  200  waits for a time T WAIT  for detection of a photon event in either detector  154  or  156 . For a high probability of an emission of from cavities  120 - 1  and/or  120 - 2 , the time T WAIT  should generally be a few times a half life Γ slow   −1 , which is the longer of the half life of excited state |e&gt; and the half life of a photon in the resonant mode of optical cavity  120 . For an N-V defect, wait time T WAIT  is preferably on the order of about 15 ns. Step  240  then waits for a further relaxation time T RELAX  for any remaining excitation in cavities  120  to relax. Relaxation time T RELAX  should typically as long or longer than time T WAIT , e.g., about 100 ns for an N-V defect. During these waits, system  100  evolves into a state |ψ 2 &gt; as shown Equation 3, where states |0&gt; i  and |1&gt; i  respectively identify states with 0 and 1 photon in mode  130 -i for i equal to 1 or 2.  
                      ψ   1     〉     ⁢     ⟶   decay     ⁢          ψ   2     〉       =       1   2     ⁡     [              ↑   ↑     〉     ⁢          0   〉     1     ⁢          0   〉     2       +            ↑   ↓     〉     ⁢          0   〉     1     ⁢          1   〉     2       +            ↓   ↑     〉     ⁢          1   〉     1     ⁢          0   〉     2       +            ↓   ↓     〉     ⁢          1   〉     1     ⁢          1   〉     2         ]               Equation   ⁢           ⁢   3             
 
         [0026]     The action of beam splitter  152  transforms state |ψ 2 &gt; of Equation 3 to a state |ψ 3 &gt; such as shown in Equation 4. In Equation 4, states |0&gt; i′ , |1&gt; i′ , and |2&gt; i′  respectively identify states with 0, 1, and 2 photons in a beam splitter output mode  135 -i for i equal to 1 or 2. As shown in Equation 4, component states of state |ψ 3 &gt; having one photon in mode  135 - 1  or one photon in mode  135 - 2  correspond to entangled states of matter systems  110 - 1  and  110 - 2 .  
                      ψ   2     〉     ⁢     ⟶   splitter   beam     ⁢          ψ   3     〉       =       1   2     ⁡     [              ↑   ↑     〉     ⁢          0   〉       1   ′       ⁢          0   〉       2   ′         +       1     2       ⁢     (            ↑   ↓     〉     +          ↓   ↑     〉       )     ⁢          1   〉       1   ′       ⁢          0   〉       2   ′         +       1     2       ⁢     (            ↑   ↓     〉     -          ↓   ↑     〉       )     ⁢          0   〉       1   ′       ⁢          1   〉       2   ′         +       1     2       ⁢          ↓   ↓     〉     ⁢     (              2   〉       1   ′       ⁢          0   〉       2   ′         -            0   〉       1   ′       ⁢          2   〉       2   ′           )         ]               Equation   ⁢           ⁢   4             
 
         [0027]     During step  230 , measurement system  150  may fail to detect any photons, register a single detection event in one of detectors  154  or  156 , or register two detection events in one of detectors  154  and  156 . However, current photomultiplier based detectors are generally unable to resolve two photons arriving in quick succession, e.g., within 10 ns, and therefore may be unable to distinguish one photon from two photons.  
         [0028]     Step  250  is a decision step that determines whether measurement system  150  has detected zero, one, or two photon events. If measurement system  150  detects zero or two photons, process  200  starts over, and step  210  again prepares the matter systems  110 - 1  and  110 - 2  in the desired initial state before entanglement process  200  resumes. On the other hand, if one (and only one) photo-detection event is observed, process  200  continues to step  260  for a second pass as described further below. If wait time T WAIT  is sufficiently long and measurement system  150  can distinguish 0, 1, or 2 photons without errors, a measurement of exactly one photon emitted heralds that matter systems  110 - 1  and  110 - 2  are in an identified one of entangled states  
              B   1     〉     =         1     2       ⁢     (            ↑   ↓     〉     +          ↓   ↑     〉       )     ⁢           ⁢   and   ⁢             ⁢             ⁢          B   2     〉       =       1     2       ⁢       (            ↑   ↓     〉     -          ↓   ↑     〉       )     .             
 
 Unitary operations can then be applied to the identified entangled state if necessary to produce a desired entangled state. 
 
         [0029]     Current light detectors are generally subject to errors including failure to detect one or more photons, falsely indicating detection of a photon (i.e., dark counts), and failure to distinguish one photon from two photons. A classical measurement error that fails to signal detection of a single photon incorrectly indicates a failure to produce an entangled state, but this type of error merely decreases the efficiency of the entanglement operation. More problematic errors occur, when two photons are detected but the classical measurement signal indicates only a single photon event or when no photon is emitted but the detector registers a dark count. For these types of errors, the desired state is not produced, even though the classical measurement signal from the first pass heralds a successful entanglement operation. Dark counts may be controlled through proper detector design and using gated operation where the detectors are only on during the short period (10 ns) when emissions are expected. Accordingly, failure to distinguish two photons from a single photon is a much more likely error for current photomultiplier based detectors. If optical measurement system  150  incorrectly identifies a 2-photon signal as a 1-photon signal, matter systems  110 - 1  and  110 - 2  may be in state |⇓⇓&gt;, even though the first heralding, i.e., the classical measurement signal, appears to indicate a successful entanglement operation. If the probability of dark counts is assumed to be negligible, a photodiode providing a first measurement result indicating a single photon results in non-coherent combination of an entangled state |B i &gt; and the state |⇓⇓&gt; as indicated by the density matrix ρ shown in Equation 5. In Equation 5, ƒ(η) indicates the probability of failure to distinguish two photons, and index i is 1 or 2 depending of which detector  154  or  156  detected the photon event.  
             ρ   =         (     1   -     f   ⁡     (   η   )         )     ⁢          B   i     〉     ⁢     〈     B   i            +       f   ⁡     (   η   )       ⁢          ↑   ↑     〉     ⁢     〈     ↑   ↑                      Equation   ⁢           ⁢   5             
 
         [0030]     After a measurement (correctly or incorrectly) indicates a successful entanglement, decision step  260  determines whether the second pass has been completed. If not, process  200  branches to step  270  and performs quantum coherent X operations on matter system  110 - 1  and  110 - 2 . An X operation, which is also referred to as a NOT operation or bit flip for a qubit, coherently interchanges state |⇑&gt; i  and state |⇓&gt; i . For the exemplary embodiment using N-V defects in diamond, the X operation can be implemented by application of an ESR pulse of the appropriate energy and duration to cause a transition between states |⇑&gt; i  and |⇓&gt; i . X operations  270  do not change the entangled states |B 1 &gt; and |B 2 &gt;, so that states of matter systems  110 - 1  and  110 - 2  remain entangled state if the measurement of a single photon during the first pass was correct. However, if the measurement during the first pass detected two photons and produced a classical measurement signal heralding a single photon, X operations  270  transform state |⇓⇓&gt; corresponding to a 2-photon signal to state |⇑⇑&gt;. Similarly, X operations  270  transform state |⇑⇑&gt; to state |⇓⇓&gt;. Process  200 , after X operation  270 , returns to step  220  and applies π-pulses to excite any components of the states of matter systems  110 - 1  and  110 - 2  corresponding to state |⇓&gt; to excited state |e&gt;. Optical measurement system  150  then detects photons emitted from cavities  120 - 1  and  120 - 2  during step  230 . If matter systems  110 - 1  and  110 - 2  are in an entangled state, measurement system  150  should again measure exactly one emitted photon. However, if the measurement during the first pass corresponded to two photons producing a single photon event, no photons are emitted during the second pass, and entanglement process  200  starts over in step  210  by initializing the states of the matter systems  110 - 1  and  110 - 2  for redoing of the first pass.  
         [0031]     Errors due to dark counts in detectors  154  and  156  can generally be controlled as noted above by restricting photon measurements to short time intervals, making the probability of dark counts low. However, some types of detectors have relatively higher probabilities of dark counts. If the first measurement signal from step  230  falsely indicated one photon when no photons where emitted, two photons should be measured during the second pass. In which case, process  200  branches from step  250  back to step  210  to restart the entanglement operation. However, the difficulty of distinguishing two photons from one photon with a photomultiplier-based detector can result in both measurements from step  230  including errors, which reduces the fidelity of entanglement process  200 . In accordance with a further aspect of the invention, immunity to a dark count errors may be further improved using third or subsequent passes where successful entanglement is heralded by detection of a single photon during every pass. For example, for a 3-pass process, process  200  branches from step  260  to step  270  after the second measurement. In the case of a dark count in the first measurement and failure to identify two photons in the second measurement, the X operations in step  270  transform matter systems  110 - 1  and  110 - 2  back to state |⇑⇑&gt;, so that the third measurement should detect no photons. Since the chance of a dark count is low, the chance of dark counts in both the first and third measurement is even lower, so that the 3-pass entanglement process may achieve even higher fidelity. Further passes may be able to further improve fidelity of the entanglement operation.  
         [0032]     After the desired number of measurements have been performed and indicate a successful entanglement, e.g., two consecutive measurements of a single photon for a double-heralded entanglement operation, decision step  260  determines that entanglement process  200  is done.  
         [0033]     Process  200  as described above can achieve high fidelity where multiple classical measurement signals heralding successful entanglement have a very low probability of being wrong. The efficiency of process  200  can still remain near a theoretic maximum 50% for the exemplary initial state. In accordance with a further aspect of the invention, an entanglement process can exceed 50%, and even approach 100% efficiency, using non-absorbing photon detectors or non-absorbing parity detectors.  
         [0034]      FIG. 3  schematically illustrates an exemplary embodiment of a quantum information system  300  in accordance with an embodiment of the invention with an optical measurement system  350  that includes non-absorbing parity detector  330 - 1  and  330 - 2 . Non-absorbing parity detectors  330 - 1  and  330 - 2  are capable of measuring the respective parities of photonic output modes  135 - 1  and  135 - 2  of beam splitter  152 . Each parity detector  330 - 1  or  330 - 2  provides a classical output signal X 1  or X 2  that indicates whether a corresponding mode  135 - 1  or  135 - 2  have even parity (i.e., a total of 0 or 2 photons) or odd parity (i.e., a total of 1 photon.)  
         [0035]      FIG. 3  shows an atypical configuration of system  300  where parity detectors  330 - 1  and  330 - 2  have different structures. The different structures are shown in  FIG. 3  primarily to illustrate some possible alternatives for the construction of parity detectors. In a more usual case, parity detectors  330 - 1  and  330 - 2  would be substantially identical to each other, e.g., both of same type as parity detector  330 - 1  or both of the same type as parity detector  330 - 2 .  
         [0036]     In the illustrated embodiment, parity detector  330 - 1  includes a controlled phase gate  331  having a phase shift constant π. A photonic probe state |α&gt;, which may be a coherent state generated by a laser or other photon source, passes through controlled phase gate  331 . Output mode  135 - 1  from beam splitter  152  controls phase gate  331 , so that controlled phase gate  331  causes a phase shift 0, π, or 2π in probe state |α&gt; depending on whether mode  135 - 1  contains zero, one, or two photons. The phase shift of 2π is equivalent to no phase shift, so that parity detector  330 - 1  cannot distinguish the 2-photon state from the 0-photon state. A phase detector  332  measures the phase shift in the probe state and produces a classical signal X 1  indicating a result of the measurement. Measurement signal X 1  indicating a phase shift with magnitude zero indicates even parity (i.e., zero or two photons) in mode  135 - 1 . A measurement X 1  indicating a phase shift with magnitude π indicates odd parity (i.e., one photon) in mode  135 - 1 . Phase gate  331  can employ a structure providing a relatively large Kerr non-linearity. Such non-linearities can be produced using electromagnetically induced transparency (EIT) such as described by W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “A High-Efficiency Quantum Non-Demolition Single Photon Number Resolving Detector”, Phys. Rev. A 71, 033819 (2005). However, in general, smaller non-linearities are easier to produce.  
         [0037]     Non-absorbing parity detector  330 - 2  illustrates a structure that can use a relatively small Kerr non-linearity to measure the parity of photons in mode  135 - 2 . As shown in  FIG. 3 , parity detector  330 - 2  includes a controlled phase gate  333 , a phase shifter  334 , and a phase shift magnitude detector  335 . Controlled phase gate  333  has a phase shift constant θ that can be much less than π. A photonic probe state |α&gt;, which may be a coherent state generated by a laser or other photon source, passes through controlled phase gate  333  and phase shifter  334  before being measured by detector  335 . Output mode  135 - 2  from beam splitter  152  controls phase gate  333 , so that controlled phase gate  333  causes a phase shift 0, θ, or 2θ in probe state |α&gt; depending on whether mode  135 - 2  contains zero, one, or two photons. Phase shifter  334  causes a fixed phase shift −θ, so that probe state |α&gt; entering detector  335  has a phase shift of −θ, 0, or θ depending on whether mode  135 - 2  contains 0, 1, or 2 photons. Phase detector  335  measures the magnitude of the phase shift in probe state |α&gt; and produces a classical signal X 2  indicating a result of the measurement. A measurement X 2  indicating a phase shift with magnitude zero indicates odd parity (i.e., one photon) in mode  135 - 2 . A measurement X 2  indicating a phase shift magnitude of |θ| indicates even parity (i.e., zero or two photons) in mode  135 - 2 .  
         [0038]     Parity detectors  3220 - 1  and  330 - 2  are non-absorbing and maintain the coherence of the photon states of modes  135 - 1  and  135 - 2 . Accordingly, photons in modes  135 - 1  and  135 - 2  pass through respective parity detectors  330 - 1  and  330 - 2  to respective input modes  342 - 1  and  342 - 2  of a 50-50 beam splitter  340 . Subsequent manipulation and measurements of the photons can be use to implement entanglement operations with high fidelity and efficiency. In the embodiment of  FIG. 3 , optical switches  352 - 1  and  352 - 2 , optical delays  354 - 1  and  354 - 2 , 50-50 beam splitters  356 - 1  and  356 - 2 , and detectors  358 - 1 ,  358 - 2 ,  358 - 3 , and  358 - 4  manipulate and measure the photons to achieve a desired entanglement of the states of matter systems  110 - 1  and  110 - 2 .  
         [0039]      FIG. 4  is a flow diagram of an entanglement process  400  that can be implemented in system  300  to efficiently entangle the states of matter systems  110 - 1  and  110 - 2 . Process  400  begins by initializing the states of matter systems  110 - 1  and  110 - 2  in a step  410  and then applying π-pulses to matter systems  110 - 1  and  110 - 2  in a step  415 . Steps  410  and  415  can be conducted in the same manner as described above for steps  210  and  220  of process  200 . For illustrative purposes, the following describes the example where the initial state of system  300  is state |ψ&gt; shown in Equation 1 above, so that the π-pulses applied in step  415  transform the initial state as shown in Equation 2. It should be understood that other initial states could also be used with similar results.  
         [0040]     After the initial state has been prepared and excited, parity detectors  330 - 1  and  330 - 2  in step  420  measure the parities of the states of respective output channels  135 - 1  and  135 - 2  of beam splitter  152 . With the exemplary initial states of matter systems  110 - 1  and  110 - 2 , system  300  just before the measurement of parities is in the state of Equation 4 above, provided that the exemplary initial states are used. The measurement results from parity detectors  330 - 1  and  330 - 2  should be “odd; even,” “even; odd,” or “even; even.” A measurement indicating “odd; odd” is not allowed because beam splitter  152  cause the photon bunching, and measurement results “odd; odd” therefore indicate an error. Accordingly, if parity detectors  330 - 1  and  330 - 2  indicate both modes  135 - 1  and  135 - 2  have odd parity, process  400  restarts by branching from decision step  425  back to step  410 . Measurements “odd; even” and “even; odd” respectively indicate entangled states |B 1 &gt; and |B 2 &gt;. Measurement “even; even” indicates that state of matter systems  110 - 1  and  110 - 2  is in the Hilbert space spanned by states |⇑⇑&gt; and |⇓⇓&gt; but further work is required to produce an entangled state.  
         [0041]     The parity measurements do not decohere the photon states, so that beam splitter  340  can undo the bunching of photons that beam splitter  152  caused. Photons from the output modes  344 - 1  and  344 - 2  of beam splitter  340  respectively enter optical switches  352 - 1  and  352 - 2 . If the measurement results from step  420  were “even; even”, a decision step  430  activates optical switches  352 - 1  and  352 - 2  to direct the measured photons into respective optical delays  354 - 1  and  354 - 2 . Such delays  354 - 1  and  354 - 2  may simply be a relatively long optical path to respective beam splitters  356 - 1  and  356 - 2 . The measured photons can be discarded for measurement results “odd; even” or “even; odd.” 
         [0042]     After step  430  or  435 , a step  440  waits the relaxation time T RELAX  before step  445  performs X operations on matter systems  110 - 1  and  110 - 2 . Step  450  then applies π-pulses to matter systems  110 - 1  and  110 - 2 , and step  455  again measures the parities of output modes  135 - 1  and  135 - 2  during a time when cavities  120 - 1  and  120 - 2  release photons produced by the decay of the excited states of matter systems  110 - 1  and  110 - 2 .  
         [0043]     Step  460  compares the parity measurements of step  420  to the parity measurements of step  455  to determine whether the measurements are consistent. Measurement results “odd; odd” in either step  420  or  455  indicate an error, causing process  400  to branch from step  460  to step  410 . If step  420  measured “odd; even” or “even; odd”, step  455  should produce the same measure results “odd; even” or “even; odd.” In either of these cases, an entangled state has been successfully produced, and step  490  can perform unitary operation on individual matter system  110 - 1  or  110 - 2  if necessary to produce the desired entangled state.  
         [0044]     If both measurement steps  420  and  455  both produce measurement result “even; even,” decision step  470  causes process  400  to branch to step  475 . During step  475  the second batch of photons, which had parities measured in step  455 , pass through detectors  330 - 1  and  330 - 2  to beam splitter  340 , from beam splitter  340  to optical switches  352 - 1  and  352 - 2 , and from optical switches  352 - 1  and  352 - 2  to input modes S 1  and S 2  of beam splitters  356 - 1  and  356 - 2 . Delays  354 - 1  and  354 - 2  provide delays such that the first batch of photons, which were produce after the first excitation of matter systems  110 - 1  and  110 - 2 , are directed toward beam splitters  356 - 1  and  356 - 2  at the same time that optical switches  352 - 1  and  352 - 2  direct the second batch of photons toward beam splitters  356 - 1  and  356 - 2 . Before the action of beam splitters  356 - 1  and  356 - 2  on the photons, system  300  will be in a state |ψ 4 &gt; given in Equation 6 if both measurement steps  420  and  455  produce results “even; even.” In Equation 6, subscripts L 1  and L 2  indicate the delayed photons and subscripts S 1  and S 2  indicate photons directly from optical switches  352 - 1  and  352 - 2 .  
                    ψ   4     〉     =       1     2       ⁡     [              ↑   ↑     〉     ⁢          1   〉       L   ⁢           ⁢   1       ⁢          1   〉       L   ⁢           ⁢   2       ⁢          0   〉       S   ⁢           ⁢   1       ⁢          0   〉       S   ⁢           ⁢   2         +            ↓   ↓     〉     ⁢          0   〉       L   ⁢           ⁢   1       ⁢          0   〉       L   ⁢           ⁢   2       ⁢          1   〉       S   ⁢           ⁢   1       ⁢          1   〉       S   ⁢           ⁢   2           ]               Equation   ⁢           ⁢   6             
 
         [0045]     Beam splitter  356 - 1  interferes photons in modes L 1  and S 1 , and beam splitter  356 - 2  interferes photons in modes L 2  and S 2 . The operation of beam splitters permit measurement of the photons in output modes dt 1  and db 1  of beam splitter  356 - 1  and output modes dt 2  and db 2  of beam splitter, and such measurements do not distinguish whether the photons were subject to the first or second set of parity measurements. Detectors  358 - 1 ,  358 - 2 ,  358 - 3 , and  358 - 4  measuring photonic modes dt 1 , db 1 , dt 2 , and db 2  identify entangled states  
              B   3     〉     =           1     2       ⁡     [            ↑   ↑     〉     +          ↓   ↓     〉       ]       ⁢           ⁢   and   ⁢             ⁢             ⁢          B   4     〉       =       1     2       ⁡     [            ↑   ↑     〉     -          ↓   ↓     〉       ]             
 
         [0046]     of matter systems  110 - 1  and  110 - 2  as indicated in Table 1. In Table 1, “Click” indicates the detector corresponding to the mode registered a photon event, and “No Click” indicates the detector corresponding to the mode did not detect a photon. Results other than those listed in Table 1 indicate a measurement error.  
                                     TABLE 1                           Entangled State Heralded by Two “Even; Even” Parity Measurements            Mode dt1   Mode db1   Mode dt2   Mode db2   Identified State               Click   No Click   Click   No Click   |B 3 &gt;       No Click   Click   No Click   Click   |B 3 &gt;       No Click   Click   Click   No Click   |B 4 &gt;       Click   No Click   No Click   Click   |B 4 &gt;                  
 
         [0047]     Process  400  as described above is thus able to produce an entangled state for all cases where parity measurements are consistent, i.e., absent an error. As a result, process  400  can approach 100% efficiency for a near deterministic entanglement operation and still provide the high fidelity resulting from double heralding.  
         [0048]     For the entanglement processes  200  and  400  described above, the dominating system imperfections that may reduce the fidelity of the heralded entangled state can be divided into three classes: (1) decoherence of matter qubits  110 - 1  and  110 - 2 ; (2) dark counts in detectors  154  and  156 ; and (3) imperfect mode matching of the photons incident on beam splitter  150 . The effect of spin decoherence of matter systems  110 - 1  and  110 - 2  depends on the way the states are generated, and can be estimated by comparing the spin decoherence time T D  with the “clock time” T C  that is provide for each quantum operation in a sequence. Clock time T C  may, for example, be about 10Γ slow   −1 . If the states of systems  110 - 1  and  110 - 2  are prepared in parallel during m clock cycles and assuming a reasonably strong cavity qubit coupling (e.g., about 100) and critically damped cavities  120 - 1  and  120 - 2 , the size of errors due to spin decoherence can be estimated to be about mT C /2T D ˜0.4γ −1 /T D , where γ is the rate of loss of photons from a cavity  120  to any mode other than modes  130 - 1  and  130 - 2  leading to measurement system  150  or  350 . For instance, for the N-V diamond system at room temperature, the cavity decay time γ −1  to loss modes is about 25 ns and decoherence time T D  is about 32 μs, making the error is about 3×10 −4     4   .  
         [0049]     A detector dark count on either round of a double-heralded entanglement operation can lead to a spurious ‘success’ of the entanglement operation, which can reduce the fidelity of the entanglement. With existing avalanche photodiodes for detectors, dark count rates are typically less than about 500 s −1 . Dark counts can be made negligible by observing the detector output during a time T WAIT  of about 3Γ slow   −1  or about 1 to 10 ns for NV-defects in diamond. The probability of a spurious dark counts for the NV-diamond system is therefore about 10 −7  to 10 −6 . Thus, dark counts in such configurations should have a negligible effect on fidelity.  
         [0050]     Imperfect mode matching of the photons emitted by the matter qubit-cavity systems reduces the fidelity because the photons carry information regarding their origin. In particular, non-identical central frequencies, different polarizations, and spatio-temporal mode shapes of the photons can all reduce the fidelity. The frequency of the photons emitted from a cavity  120 -i depends on the photon frequency ω et  associated with the transition of the excited state and the cavity mode frequencies ω ci  associated with cavity  120 -i. The transition frequency ω et  can be tuned independently, e.g., using local electric and magnetic fields to induce Stark and Zeeman shifts. The cavity mode frequencies ω ci  can also be accurately and independently tuned, e.g., using strain-tunable silica microcavities or piezoelectrically tuned fiber optic microcavities. The polarization of the emitted photons can be accurately matched using linear optical elements such as polarization rotators calibrated so that photons emitted from any cavity have the same polarization. The spatio-temporal mode shapes of the emitted photons depend on the couplings and the damping of the respective cavities. The coupling and damping parameters of a cavity in general, depend on the structure of the cavities, and hence are more difficult to calibrate once the cavities have been fabricated. However, it can be shown that the entanglement operation is rather robust to mismatches in the coupling and the damping parameters of cavities  120  and that mismatches of a few percent in the cavity parameters lead to a reduction in fidelity of less than 10 −3 .  
         [0051]     In accordance with a further aspect of the invention, entanglement operations such as described above can be used to link qubits together into cluster states, for possible uses in scalable quantum computers.  FIG. 5  shows an N-qubit system  500  suitable for construction cluster states. System  500  includes N matter systems  110 - 1  to  110 -N, which are enclosed in respective cavities  120 - 1  to  120 -N. As described above, each matter system  110 -i has states |⇑&gt; i  and |⇓&gt; i  that correspond to the basis states of the ith qubit and an excited state |e&gt; i  that couples solely to state |⇓&gt; i  through single photon emission. In accordance with a principle of the present invention, matter systems  110  can be widely separated in a distributed quantum information system that uses optical fibers or other techniques to convey photons from the remote matter systems  110 . Alternatively, some or all of matter systems  110  can be local and/or integrated into a single device.  
         [0052]     Excitation system  160  can selectively drive any matter system  110 -i with a π pulse corresponding to the transition from state |⇓&gt; i  to state |e&gt; i . Excitation system  160  may also be used to apply electromagnetic or other types pulses (e.g., ESR pulses) for single-qubit operations. Although shown as a single block in  FIG. 5 , excitation system  160  may be implemented using multiple separate systems, particularly when matter systems  110  are separate parts of a distributed quantum information system.  
         [0053]     In the embodiment illustrated in  FIG. 5 , an optical switch or network  530  is positioned to receive photons that leak from cavities  120 - 1  to  120 -N and to route output modes from any pair of cavities  120 - 1  to  120 -N to measurement system  550 . Measurement system  550  can be similar or identical to measurement system  150  or  350  as described above in regard to  FIGS. 1 and 3 . System  500  can thus perform an entanglement operation (e.g., process  200  of  FIG. 2  or process  400  of  FIG. 4 ) on that states of any pair of matter systems  110 -i and  110 -j (for i≠j) that optical switch  530  selects. Alternative structures could also be employed for selection of matter systems  110 -i and  110 -j. For example, a generalized multi-port beam splitter can implement both qubit selection and interfere emitted photons for the required measurements.  
         [0054]     System  500  can construct a cluster state by entangling the states of a pair of matter systems  110  when the state of at least one of the matter systems  110  in the pair is already entangled with the state of another matter system  110 . In general, a one-dimensional cluster |C&gt; 1 . . . N  of N qubits (a chain) may be represented in the form of Equation 7 where Zi represents the Pauli phase-flip operation acting on qubit i.
 
| C&gt;   1 . . . N =(|⇑&gt; 1 +|⇓&gt; 1   Z   2 )(|⇑&gt; 2 +|⇓&gt; 2   Z   3 ) . . . (|⇑&gt; N +|⇓&gt; N )  Equation 7
 
         [0055]     In accordance with an aspect of the invention, a process  600  such as illustrated in the flow diagram of  FIG. 6  can generate N-qubit o cluster state |C&gt; 1 . . . N  from an (N- 1 )-qubit cluster |C&gt; 2 . . . N  by entangling a qubit with a qubit in cluster state |C&gt; 2 . . . N . Process  600  starts in steps  610  and  615  with the preparation of cluster state |C&gt; 2 . . . N  and a single-qubit in an initial state. Cluster state |C&gt; 2 . . . N  can be prepared by repeated use of process  600  starting from single qubits. The initial state |+&gt; 1  will be used in the following example, but other initial states could be employed. In the manner described above, πpulses are applied to matter systems, e.g.,  110 - 1  and  110 - 2 , respectively corresponding to the single qubit state |+&gt; 1  and the qubit at an end of the chain of qubits in cluster state |C&gt; 2 . . . N . Steps  630  and  640  then measure photons produced during time T WAIT  and then wait a relaxation time T RELAX . The measurement during step  630  can be photon detection or parity measurements depending on the type of entanglement operation. Decision step  650  determines whether the first measurements indicate the entanglement process should proceed. If not, the entanglement operation failed, and process  600  branches to end step  655 , which has the consequences described further below. If the first measurements indicate process  600  should proceed, decision step  260  begins a second pass by directing process  600  to step  670 , which performs bit flip operations on matter systems involved in the entanglement operation. Steps  620 ,  630 , and  640  are then repeated, and step  650  again determines whether the entanglement operation has failed.  
         [0056]     If the entanglement operation is successful, the resulting state of matter systems  110 - 1  to  110 -N will be one of the states |χ 1 &gt;, |χ 2 &gt;, |χ 3 &gt;, and |χ 4 &gt; of Equations 8 when step  660  determines that the second pass is complete. The measurement results during the two repetitions of step  630  specifically indicate which state |χ 1 &gt;, |χ 2 &gt;, |χ 3 &gt;, or |χ 4 &gt; was produced. Step  680  can then perform local transformations on qubits 1 and 2 to produce the desired cluster state. Using entanglement process  200  in process  600  produces either state |χ 1 &gt; or |χ 2 &gt;, depending on which of detector  154  or  156  detected the photon during the last measurement step  630 . Using entanglement process  400  in process  600  produces states |χ 1 &gt; and |χ 2 &gt; when the results of the parity measurements are respectively “odd; even” and “even; odd.” States |χ 1 &gt; and |χ 2 &gt; can be transformed into a one-dimensional cluster state by applying the respective local operations H 1 X 2  and X 1 H 1 X 2 , where H i  is a Hadamard operation on the ith matter qubit, and X i  the Pauli operator implementing a bit flip on the ith qubit. When the parity measurement results during entanglement process  400  are both “even; even”, state |χ 3 &gt; or |χ 4 &gt; is produced depending on the measurement results from detectors  358 - 1  to  358 - 4 . (See Table 1.) State |χ 3 &gt; can be converted to a one-dimensional cluster state through application of local Hadamard operation H 1 , and state |χ 4 &gt; can be converted to a one-dimensional cluster state through application of local operation X 1 H 1 .  
                          χ   1     〉     =       (              ↑   〉     1     ⁢          ↓   〉     2       +            ↓   〉     1     ⁢          ↑   〉     2     ⁢     Z   3         )     ⁢          C   〉       3   ⁢           ⁢   …   ⁢           ⁢   N           ⁢     
     ⁢            χ   2     〉     =       (              ↑   〉     1     ⁢          ↓   〉     2       -            ↓   〉     1     ⁢          ↑   〉     2     ⁢     Z   3         )     ⁢          C   〉       3   ⁢           ⁢   …   ⁢           ⁢   N           ⁢     
     ⁢            χ   3     〉     =       (              ↑   〉     1     ⁢          ↑   〉     2       +            ↓   〉     1     ⁢          ↓   〉     2     ⁢     Z   3         )     ⁢          C   〉       3   ⁢           ⁢   …   ⁢           ⁢   N           ⁢     
     ⁢          χ   4         〉     =       (              ↑   〉     1     ⁢          ↑   〉     2       -            ↓   〉     1     ⁢          ↓   〉     2     ⁢     Z   3         )     ⁢          C   〉       3   ⁢           ⁢   …   ⁢           ⁢   N                 Equations   ⁢           ⁢   8             
 
         [0057]     If the entanglement operation fails, i.e., process  600  branches to step  655 , and in general, the state of qubit 2 is unknown. However, measuring qubit 2 in the computational basis removes qubit 2 from the cluster and projects qubits {3 . . . N} back into a pure cluster state |C&gt; 3 . . . N . Therefore, failure of operation  600  causes the original cluster to shrink by one qubit.  
         [0058]     Repeatedly applying process  600  as described above allows long chains to be grown. However, the theoretical upper limit on the success probability p of process  600  using entanglement process  200  is ½, and when the entanglement operation fails, the chain shrinks by 1 qubit. Therefore, process  600  using entanglement process  200  alone cannot create large clusters efficiently. If recycling of the cluster state after a failure of an entanglement operation is not performed, the average number of entanglement operations required to create a cluster of m qubits is  
           N   EO     =         ∑     i   =   1       m   -   1       ⁢     p     -   i         =         p     1   -   m       ⁡     (     1   -     p     m   -   2         )       /     (     1   -   p     )           ,       
 
 where p is the probability of the entanglement operation succeeding. Accordingly, the average number N EO  of entanglement operations required grows exponentially with the number m of qubits in the cluster. Even using entanglement operation  400  of  FIG. 4 , which has an efficiency p that is not theoretically bounded to ½ but can approach 1, may require a large number of entanglement operations to produce a large cluster state. 
 
         [0059]     In accordance with a further aspect of the invention, large cluster states can be more efficiency generated by growing a cluster state up to a critical size that depends on entanglement success probability p and then using the entanglement operation together with local operations to join the cluster state having the critical length to a cluster state being lengthened. In particular,  FIG. 7  is a flow diagram of a process  700  for producing longer one-dimensional cluster states. Process  700  begins with steps  710  and  720  of producing first cluster state of length N to be lengthened and a second cluster state of length m that is at least of the critical length. The first and second cluster states can be produced using process  600  of  FIG. 6  or process  700  of  FIG. 7 . The state of a system including sets of qubits in the two one-dimensional cluster states may be written as shown in Equation 9. In Equation 9, states |C&gt; {A}  and |C&gt; {B}  respectively represents one-dimensional cluster states of qubits {A2 . . . AN} and {B2. . . Bm}.
 
(| C&gt;   {A} |⇑&gt; A1   +Z   A2   |C&gt;   {A} |⇓&gt; A1 )(| C&gt;   {B} |⇑&gt; B1   +Z   B2    |C&gt;   {B} |⇓&gt; B1 )  Equation 9
 
         [0060]     The first and second cluster states |C&gt; {A}  and |C&gt; {B}  can be joined by first performing a local operation X A1  on an end qubit of cluster state |C&gt; {A}  in step  730  and then in step  740  performing an entanglement operation between qubit A 1  and an end qubit B 1  of cluster state |C&gt; {B} . If step  750  determines the entanglement operation is successful, step  760  measures qubit B 1  in a rotated basis |±&gt; )B1 =|⇓&gt; B1 ±|⇑&gt; B1 , and the qubits remaining after measurement  760  are left in a state of Equation 10. In Equation 10, the first sign depends on the outcome of the measurement of qubit B 1 , and the second sign depends on the measurement results during the entanglement operation. Step  770  performs a local operation on qubit A 1  yields to transform the unmeasured qubits to a cluster state of qubits {AN, . . . A 1 , B 2 , . . . Bm} having length N+m−1. If the entanglement operation fails, qubit A 1  can be measured in the computational basis, and the original cluster state shrinks by 1 qubit. Thus, the average length L of the new cluster is p(N+m−1)+(1−p)(N−1). In order that the cluster grows on average, the average length L must be greater than the length N of the cluster state being extended, which implies that length m of the other cluster state should be greater than a critical length 1/p.
 
±|⇑&gt; A1   |C&gt;   {A}   |C&gt;   {B} ±|⇓&gt; A1   Z   A2   |C&gt;   {A}   |C&gt;   {B}   Equation 10
 
         [0061]     Chains of fixed length m can be grown independently by sequentially adding single qubits as in process  600  of  FIG. 6  or by the end-to-end joining of multi-qubit clusters. Growing these m-qubit chains adds a constant overhead cost to the cluster growth process. For example, growing a 4-qubit chain (without recycling) requires on average p −3 +p −2 +p −1  attempted entanglement operations, and each attempt to join such a chain adds on average 4p−1 qubits to the large cluster, leading to a fixed total cost of C 4 =(p −3 +p −2 +p −1 +1)/(4p−1) entanglement operations per qubit added to the large cluster. Joining two 3-chains together can grow a 5-chain. Joining such 5-chains to a longer cluster leads to a total cost of C 5 =(2p −3 +2p −2 +p −1 +1)/(5p−1) entanglement operations per qubit.  
         [0062]     To minimize these costs, the collection and detection efficiencies of the system should be maximized. However, process  700  can successfully grow cluster states even when the probability is small, e.g., less than ¼. For example, for probability p about equal to 0.24 (or η=70% with γ=0), the critical length mc is 5, and the average cost C 5  to produce a one-dimensional cluster state of length 5 is 775 entanglement operations. A modest improvement in detector efficiency dramatically reduces the overhead cost: for η=85% and γ=0, we find C 4 =73.4. Efficiency for growing clusters may be improved using entanglement operations and recycling of small clusters to lower overhead costs.  
         [0063]     One-dimensional clusters are generally not sufficient for simulating arbitrary logic networks, and therefore it is desirable to be able to generate more general graph states. In accordance with an aspect of the invention, the entanglement operations disclosed above can be used for creating vertical bonds between cluster states and for constructing two-dimensional or higher-dimensional cluster states. One such technique for combining one-dimensional cluster states to form two-dimensional cluster states capable of simulating arbitrary logic networks begins by constructing two one-dimensional cluster states (e.g., using the techniques described above) and then entangling a qubit that is from the first one-dimensional cluster but is not at the end of the first one-dimensional cluster state with a qubit that is from the second one-dimensional cluster state but not at the end of the second one-dimensional cluster state. If the entanglement operation is successful, single-qubit operations can then be used to produce a cluster state corresponding to an I-shaped structure. Further, qubits, one-dimensional cluster states, or higher-dimensional cluster states can then be attached using entanglement operations on target qubits.  
         [0064]     Other techniques for using entanglement operations to construct higher-dimensional cluster states are known and could employ the entanglement operations in accordance with the present invention. For example, Browne and Rudolph, Phys. Rev. Lett. 95, 010501 (2005) describes a technique using measurements to convert a qubit in linear chain to a dangling bond or “cherry” that can be entangled with other “dangling bond” qubits to create vertical bonds between linear chains. Such techniques have the advantage that a failed entanglement operation does not break the linear chains in two, but only shortens the chains.  
         [0065]     Some embodiments of the invention as described above can provide very desirable features. First, the matter systems require only a simple level structure and single-qubit operations. Second, photon loss does not reduce the fidelity of the entangled state of the qubits, but merely adds to the constant overhead cost. Third, owing to the simplicity of the optical networks, mode matching is relatively straightforward. Fourth, the entanglement operation is inherently distributed so that individual qubit-cavity systems can be placed at distant sites, and connected by optical fibers. Accordingly, embodiments of the invention can be used in distributed applications such as quantum repeaters and quantum cryptography.  
         [0066]     Although the invention has been described with reference to particular embodiments, the description is only an example of the invention&#39;s application and should not be taken as a limitation. Various adaptations and combinations of features of the embodiments disclosed are within the scope of the invention as defined by the following claims.