Abstract:
The invention provides a system and method for quantum computation comprising the steps of defining a Hilbert space to represent a physical quantum mechanical system; selecting a set of quantum observables for said system; selecting a subspace of said Hilbert space in which all states representing said quantum mechanical system have the same eigenvalue; and storing information in or processing information using said subspace. In addition, the invention provides a system and method for quantum computation, comprising defining a Hilbert space to represent a physical quantum mechanical system; selecting a subspace of said Hilbert space in which the quantum mechanical state of the physical system is decoherence free; and processing quantum information using the quantum mechanical system by controlling only multi body interactions.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application claims priority to U.S. provisional application No. 60/283,016, filed Apr. 11, 2001. The contents of that provisional application are incorporated herein by reference. 
     
    
     
       DISCUSSION OF THE BACKGROUND  
         [0002]    1. Field of the Invention  
           [0003]    This invention relates to computation, and more specifically to quantum cryptography and quantum computing.  
           [0004]    2. Discussion of the Background  
           [0005]    Quantum mechanical systems have been investigated for many applications including quantum cryptography and quantum computation. Information may be stored and processed in such a quantum system. Often the information is carried by two-state quantum subsystems. Each two-state quantum subsystem is called a quantum bit (“qubit”). Such quantum mechanical processing systems may outperform classical computers in such tasks as prime factorization problems.  
           [0006]    Physical implementations of qubits have the problem that the quantum system which carries the information are coupled to the environment. This leads to a process known as decoherence in which encoded quantum information is lost to the environment. U.S. Pat. No. 6,128,764 to Gottesman discusses quantum cryptography and quantum computation.  
           [0007]    In order to remedy this problem, active quantum error-correcting codes (QECCs) have been developed in analogy with classical error correction. These codes encode quantum information over an entangled set of code words the structure of which serves to preserve the quantum information when frequent measurements of the errors and correction of any resulting errors are done.  
           [0008]    One major development in the formulation of QECC has been the advent of stabilizer code theory. In this formalism, a set of stabilizer operators is defined, and the code is the common eigenspace of the stabilizer elements with eigenvalue +1.  
           [0009]    The decoherence-free subsystem (DFS) approach is another method that has been developed to combat decoherence resulting from a specific decoherence mechanism. In contrast to the active mode of QECC, the DFS approach utilizes the symmetry of the system-environment coupling to find a subspace of the system which does not experience decoherence. The quantum information the user wishes to encode in the system is expressed by means of states contained in the decoherence-free subsystem.  
           [0010]    The inventors recognized that, from the perspective of quantum computation, it is also important to be able to controllably transform decoherence-protected states. Once the information encoded in the quantum system has been protected from decoherence by some method such as a DFS, it is useful to be able to perform universal quantum computation. The idea of universal computation is the following: with a restricted set of operations called a universal set, one wishes to implement any unitary transformation on the given Hilbert space to an arbitrary degree of accuracy. There are a number of universal sets that have been identified.  
         SUMMARY OF THE INVENTION  
         [0011]    It is an object of the invention to provide improved quantum computation.  
           [0012]    It is another object of the invention to provide improved decoherence free quantum computation.  
           [0013]    These and other objects of the invention are provided by a system and method for quantum computation comprising the steps of defining a Hilbert space to represent a physical quantum mechanical system; selecting a set of quantum observables for said system; selecting a subspace of said Hilbert space in which all states representing said quantum mechanical system have the same eigenvalue; and storing information in or processing information using said subspace.  
           [0014]    The objects are also provided by a system and method for quantum computation comprising defining a Hilbert space to represent a physical quantum mechanical system; selecting a subspace of said Hilbert space in which the quantum mechanical state of the physical system is decoherence free; and processing quantum information using the quantum mechanical system by controlling only multi body interactions.  
           [0015]    In accordance with the present invention, an apparatus and method are described for forming a decoherence-free subsystem based on finding a set of quantum observables. These quantum observables are determined based on the primary mechanism for decoherence in the system. The Hilbert space is then divided up into subspaces according to the eigenvalues of these observables. A specific simultaneous subspace with eigenvalue +1 for all said quantum observables is selected as the decoherence free subsystem.  
           [0016]    Physical systems for implementing quantum computing include NMR machines, quantum optics, superconducting electronics, quantum-dot physics, electrons flowing on Helium, and silicon systems. See for example S. L. Braunstein and H. -K. Lo, “Scalable quantum computers, paving the way to realization”, Wiley-VCH, 2001, the teachings of which are incorporated herein by reference.(See http://www.wiley-vch.de/books/tis/eng/3-527-40321-3.html).  
           [0017]    The invention also provides an apparatus and method for performing universal quantum computation on the DFS by controlling only multi-body interactions in the physical system.  
           [0018]    Stabilizers are used to identify a set of Hamiltonians which are composed to find a universal set of gates on the DFS-encoded system. This universal set is then used to approximate any gate on the encoded system; these gates have the property that when applied to a state in the DFS, the state never leaves during the entire time that the gate is applied.  
           [0019]    The invention further provides an apparatus and method for performing universal quantum computation under a particular ubiquitous form of decoherence mechanism called strong collective decoherence (SCD), in which stabilizers are used as above to derive a set of Harniltonians called two-body exchange operators. Arbitrary gates acting on the DFS states are constructed using these exchange operators such that the states never leave the DFS during the entire process of applying the gate.  
           [0020]    A preferred example of an apparatus forming a physical embodiment of a decoherence-free subsystem, for performing universal quantum computation on the DFS, and for performing universal quantum computation using strong collective decoherence (SCD) is a nuclear magnetic resonance apparatus enclosing a molecule wherein the molecule has n magnetically distinct spin states, and a controller for providing predetermined sequences of pulse of various frequencies, power levels, and durations intended to excite the nuclei of the molecule into any state inside the 2 ** n dimensional Hilbert space that describes the state of the physical system. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0021]    The invention will be described with reference to the following drawings.  
         [0022]    [0022]FIG. 1 is a schematic showing a NMR system for performing universal quantum computation on the DFS, and SCD.  
         [0023]    [0023]FIG. 2 is a flowchart showing operation steps in forming a decoherence-free subsystem based on stabilizer formalism; and  
         [0024]    [0024]FIG. 3 is a flowchart showing construction of an arbitrary gate acting on the decoherence-free subsystem that never takes states out of the said subsystem. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0025]    [0025]FIG. 1 is a schematic showing a NMR system for performing universal quantum computation on the DFS, and SCD.  
         [0026]    [0026]FIG. 1 shows NMR apparatus  1  including NMR sample holder  10 , sample  20 , Bo field coils  30 , B1 field coils  40   a ,  40   b , signal transmission line  50 , input/output coupler  60 , amplifier  70 , mixer  80 , local oscillator  90 , power mixer  100 , pulse train signal  110 , and control circuitry  120 .  
         [0027]    In operation, sample  20  contains a molecule (or a large number of the same molecule) having N magnetically distinct nuclear spins. Each spin may be of spin-½ and represents a qubit (i.e., quantum bits). Therefore, the N nuclear spins represent N qubits. The wave function of the N qubits, therefore, lives (or is defined in) in a 2**n dimensional Hilbert spaces. DC magnetic field Bo is applied by coil  30 . Control circuitry  120  generates pulse train signal  110 . Pulses train signal  110  is converted by power mixer  100  to a signal of suitable frequency and power to provide induce current in coils  40   a ,  40   b . The current induced in coils  40   a ,  40   b  generates a magnetic field at the location of the molecule. The nuclear spin of the molecule in sample  20  reacts to the magnetic field wave form by changing state. The change in state of the molecule may be detected by measuring power transmitted back from coils  40   a ,  40   b , through coupler  60 , amplifier  70 , and mixer  80  to control circuitry  120 . In essence, the n nuclear spins of the molecule can be used as a storage device for quantum information that represents a wave function living in a 2**n dimensional physical Hilbert space. (In an actual NMR quantum computing, labeling (also referred to as encoding) may reduce the actual number of *computational* qubits that are represented by the n physical nuclear spins in a molecule. See Braunstein and Lo, supra, for details.) The spin state of the molecule as described above is analogous to an n-bit register that can represent an n-bit number (i.e., an element in a finite field in number theory) in a conventional computer, which is a physical system. In quantum computing, controlled interactions acting on the n nuclear spin can lead to a corresponding controlled quantum logical operation acting on the wave function of the 2**n dimensional Hilbert space. This is somewhat analogous to a conventional digital computer where controlled physical interactions will lead to a controlled logical operation on the state of the n-bit register. The main difference is that quantum information lives in (or is defined with respect to) a Hilbert space, which is a much larger space than the space of n bits. Therefore, quantum computing with a physical system of n qubits offers more computing power than conventional digital computing with a physical system of similar physical size, namely, n bits.  
         [0028]    The theory and experiment of NMR quantum computing has recently been discussed in scientific papers. See, for example, S. L. Braunstein and H. -K. Lo, “Scalable quantum computers, paving the way to realization”, Wiley-VCH, 2001.  
         [0029]    The decoherence-free subsystem presented in this disclosure can be modeled as a system S coupled to a bath B. The system will represent the quantum device in which information is to be stored and processed, while the bath represents the environment to which the quantum device is coupled. The interaction between the system and the bath is characterized by the Hamiltonian H=H S {circle over (x)}I B +I S {circle over (x)}H B +H I . Here H S (H B ) is the system (bath) Hamiltonian that acts on the system (bath) Hilbert space H S (H B ), I S (I B ) is the identity operator on the system (bath) Hilbert space, and H I  is the interaction Hamiltonian containing all the nontrivial couplings between system and bath. This interaction Hamiltonian can be written generally as the sum of operations which act separately on the system (S α ) and on the bath (B α ): H I =ΣS α B α . This interaction Hamiltonian, over time, mixes information encoded in the system S with states in the bath B; thus monitoring of the system shows a loss of information. To reflect this, the S α  will be referred to as the error operators on the system. This leakage of quantum information from the system to the bath is the decoherence process against which a DFS will protect.  
         [0030]    Stabilizer Formalism Stuff.  
         [0031]    [0031]FIG. 2 is a flowchart showing the primary operation steps in forming such a decoherence-free subspace using the method of stabilizer codes. At step  210 , a set of quantum observables is selected, as will be explained subsequently, to be the stabilizer group that acts on quantum states in the Hilbert space of the quantum system under consideration. At step  220 , the Hilbert space is divided up into subspaces according to the eigenvalues of the quantum observables selected in step  210 , and a specific subspace corresponding to the eigenspace in which all the elements of the stabilizer group simultaneously have eigenvalue +1 is chosen to be the decoherence-free subsystem (DCF). Finally, in step  230 , information is stored or processed in such a DCF subsystem, by means of a set of states encoding the information which are chosen so that they lie in the decoherence-free subsystem. The fact that the information is in a DCF subsystem defined by  220  means that the information is protected against any errors generated by the stabilizer group found in  210 , and in addition can detect error operators that do not commute with at least one of that stabilizer group.  
         [0032]    It is important to be able to choose the stabilizer group in step  210  given a particular set of error operators {S α }. This selection is done by noting some features of a DFS. A DFS is defined as a subsystem of the quantum system under consideration such that the system evolves unitarily under the error operators. It is possible to show that a DFS is completely specified by the following condition: |u&gt;{circle over (x)}|λ&gt; is in the DFS if and only if S α |μ&gt;{circle over (x)}|λ&gt;=|μ&gt;{circle over (x)}M α |λ&gt; (eqn1). The DFS stabilizer D(v) which stabilizes the DFS with eigenvalue +1 is  
         D        (     v   ρ     )       =       D        (       v   1     ,   …              ,     v   N       )       =     exp              [       ∑   α            v   α          (       S   α     -     I   ⊗     M   α         )         ]                             
 
         [0033]    for all complex v α .  
         [0034]    The following will show that D(v) is the DFS stabilizer. The DFS condition implies that the D(v) stabilize the DFS:  
         D                     (   v   )     ρ                   μ     〉     ⊗        λ       〉       =            μ     〉     ⊗            λ     〉     .                             
 
         [0035]    Moreover,  
         D                     (   v   )     ρ                   μ     〉     ⊗        λ       〉       =              μ     〉     ⊗        λ       〉                           
 
         [0036]    for all complex v implies that |μ&gt;{circle over (x)}|λ&gt; is in the DFS. This is because it follows from the definition of the above stabilizer that D(0,0, . . . 0, v β ,0, . . . ,0)|μ&gt;{circle over (x)}|λ&gt;=|μ&gt;{circle over (x)}|λ&gt;. By picking v β  to be very small, it follows that  
           D                     (   v   )     ρ               μ              〉     ⊗        λ   〉         -          μ              〉     ⊗        λ   〉         =         v   β          (       S   β     -     I   ⊗     M   β         )                 μ   〉     ⊗        λ   〉                               
 
         [0037]    up to some additive factor which can be made as small as wished, which shows that D(v) is the stabilizer which stabilizes the DFS given by the condition (eqn1). Therefore, we can pick such a stabilizer set as above, and as in step  220 , a state belongs to the DFS if and only if the state belongs to the simultaneous eigenspace of the stabilizer set with eigenvalue +1.  
         [0038]    Universal Quantum Computation  
         [0039]    One result of constructing a DFS using the stabilizer method is that it helps elucidate a protocol for performing information processing on the DFS-encoded states as in  230 . It is wished to construct arbitrary gates for which which the states upon which the gates act stay within the DFS at all times, so that the performance of the gate does not result in decohering the state.  
         [0040]    [0040]FIG. 1 shows a quantum device for information processing that implements universal computation in accordance with the method just describe with reference to FIG. 2. The molecule is the quantum medium comprised of a quantum system and its surrounding environment, or bath, where the quantum system has many subsystems with at least two quantum states each. The NMR apparatus surrounding the molecule/medium controls the quantum states in the medium by applying the gates as per steps  220 ,  230 .  
         [0041]    [0041]FIG. 3 is a flow chart showing steps for approximating any gates on encoded states of the system.  
         [0042]    In step  310 , the stabilizer set is used to identify a set of Hamiltonians {H i } such that these Hamiltonians, when acting on a state in the DFS for a specified time, keep that state within the DFS for the entire switching time of the gate. In step  320 , the {H i } are composed to find a universal set of gates on the DFS-encoded states. Finally, in step  330 , the universal set is used to approximate any gate on the encoded states.  
         [0043]    The procedure used to approximate any gate on the encoded states is to note that if a unitary gate U takes states in the DFS to other states in the DFS,  
           U      D                       (   v   )     ρ          U   +       =     D                     (     v   ′     )     ρ                             
 
         [0044]    for all v. Then the condition that U and D commute, UD=DU, is certainly a sufficient condition for the unitary gate U to preserve the DFS. These unitaries are physically implemented by turning on a Hamiltonian H for a certain time t: U(t)=e iHt . The stabilizer set is further important to identify the more restrictive set of Hamiltonians that not only preserve the DFS at the conclusion of the gate operation, as do the commuting U&#39;s above, but also preserve the DFS throughout the entire duration of the gate operation. Taking the derivative of U(t)D(v)=D(v)U(t) with respect to time and evaluating the resulting equation at t=0 gives a sufficient condition to identify the Hamiltonians of interest in step  202 : the Hamiltonians H such that HD=DH.  
         [0045]    Universal Quantum Computation Inside SCD-DFS  
         [0046]    An important design consideration is the choice for the quantum medium, such as the molecule and its nuclear spin states in FIG. 1. It is important for the quantum medium to have the right kind of decoherence mechanism in order to implement the DFS-stabilizer construction in FIGS. 2 and 3. This mechanism should take the form of collective decoherence, in which there is some symmetry of the interaction with the environment.  
         [0047]    A particularly useful decoherence mechanism for the quantum medium is one where the bath cannot distinguish the system qubits. The interaction Hamiltonian describing this mechanism is most easily expressed by defining the Pauli operators  
           σ   x     =     (         0       1           1       0         )       ,                  σ   y     =     (         0         -   i             i       0         )       ,                  σ   z     =     (         1       0           0         -   1           )                             
 
         [0048]    in the basis spanned by σ z  eigenstates |0&gt; and |1&gt;. It is now possible to define the collective operators  
         S   α     ≡       ∑     j   =   1     n            σ   α   j     .                             
 
         [0049]    Here σ α   j  denotes a tensor product of the identity on all but the jth qubit and the αth Pauli matrix on the jth qubit, where α=x,y,z. For n qubits interacting with a bath, the most general interaction Hamiltonian linear in the σ α   j  can be written as  
           H   I     =       ∑     i   =   1     n            ∑       α   =   x     ,   y   ,   z              σ   α   i     ⊗     B     i   ,   α               ,                         
 
         [0050]    where the B i,α  are bath operators. If the bath cannot distinguish the system qubits, B i,α  should not depend on i. Then the interaction Hamiltonian for the case in which the bath cannot distinguish the system qubits can be written as  
         H   I     =       ∑       α   =   x     ,   y   ,   z              σ   α   i     ⊗       B   α     .                               
 
         [0051]    Collective decoherence mechanisms for which this interaction Hamiltonian holds will be called strong collective decoherence or SCD.  
         [0052]    The situation in which the bath is the electromagnetic field approximates this requirement, where the wavelength of the transition between the states of the qubits is larger than the spacing between the qubits. The electromagnetic field will then interact with each of these qubits in an identical manner, because the field strength over a single wavelength will not vary substantially. For condensed-phase -&gt; 
         [0053]    SCD is a useful decoherence mechanism because of the particular form of the error operators. The operators {S α } acting on n qubits are mathematically equivalent to the total angular momentum operator for n spins that is frequently encountered in quantum mechanics; therefore, the same formalism used to describe angular momentum can be used in this case. This formalism says that the Hilbert space of n qubits can be decomposed into subsystems which correspond to different values of the total angular momentum, and states can be described in terms of their total angular momentum, J; the degeneracy of the decomposition for a given J, λ; and the J z  value within the (J,λ)-subsystem, μ. Such a state will therefore be described as |J, λ, μ&gt;. From the theory of angular momentum it is easy to see that the error operators act as the identity on the λ component, so a DFS is defined by fixing J and μ. The DFSs corresponding to the different J values for a given n can be computed using standard methods for the addition of angular momentum. The convention used here will be that |1&gt; represents a |j=½, mj=½&gt; particle and |0&gt; represents a |j=1/2, mj=−1/2&gt; particle, though the labeling does not represent a physical angular momentum, as the {S α } are not actually angular momentum operators.  
         [0054]    The smallest n that supports a DFS and encodes at least a qubit of information is n=3. For this case there are two possible values of the total angular momentum: J={fraction (3/2)} or J=½. The J=½ states have degeneracy 2. This degeneracy results from two ways to construct the J=1/2 state by adding angular momentum of three spins: one can add either a J=1 or J=0 state on the first two spins to the J=½ state on the third spin. The four J=½ states can encode one qubit through this degeneracy; the resulting DFS on three qubits will be denoted as DFS 3 (½). In the following equation, this encoding will be shown in several steps:  
                      0   L     〉     =     {                        1   2     ,   0   ,   0       〉     =                      0   ,   0       〉     ⊗            1   /   2     ,       -   1     /   2           〉     =       1     2            (        010           〉     -        100         〉                            1   2     ,   0   ,   1       〉     =                      0   ,   0       〉     ⊗            1   /   2     ,       +   1     /   2           〉     =       1     2            (        011           〉     -        101         〉                               □                      1   L     〉     =     {                        1   2     ,   1   ,   0       〉     =                                   1     3            (       -     2                 1   ,     -   1               〉     ⊗            1   /   2     ,     1   /   2           〉     +          0   ,   0         〉     ⊗            1   /   2     ,       -   1     /   2           〉     )     =       1     6            (       -   2             001             〉     +        010       〉     +        100       〉       )                                                    1   2     ,   1   ,   1       〉     =           1     3            (       -     2                 1   ,     +   1               〉     ⊗            1   /   2     ,       -   1     /   2             〉     -          1   ,   0         〉     ⊗            1   /   2     ,       +   1     /   2           〉     )     =       1     6            (     2           001             〉     +        010       〉     +        100       〉     )                                                 
 
         [0055]    (eqn)  
         [0056]    In this equation, the first column indicates the grouping forming a logical qubit; the second uses the |J, λ, μ&gt; notation; the third uses tensor products of the state on the first two spins with the state on the third spin; and the fourth expands the states in terms of the single-particle |j=½, mj=±½&gt; basis using Clebsch-Gordon coefficients.  
         [0057]    SCD is also a mechanism for which the universal computation scheme of FIG. 2 can be exhibited using only two-body exchange Hamiltonians. If the system is expressed in the basis |J 1 , . . . ,J n-1 ,J;m J &gt;, where the J i  express the angular momentum of the first i qubits, it can be shown from angular momentum formalism considerations that the errors {S α } act only on the m J  component: S α |J 1 , . . . ,J;m J &gt;=|J 1 , . . . ,J n−1 ,J&gt;{circle over (x)}P α |m θ &gt;, where the P α  denote operators acting directly on the |m θ &gt; component of the DFS. Then the stabilizer for this system is  
         D        (       v   x     ,     v   y     ,     v   z       )       =       exp              [       ∑       α   =   x     ,   y   ,   z              v   α          (       S   α     -     I   ⊗     P   n         )         ]     .                           
 
         [0058]    The Hermitian two-body exchange interactions, which flip qubits i and j and leaves other states invariant, commute with the stabilizer: E ij =|k&gt; i |l| j =|l&gt; i |k&gt; j . They thus preserve the SCD DFS, and indeed are the operators which act on the smallest number of qubits that also preserve the DFS.  
         [0059]    The following will examine whether these two-body exchange operators are sufficient for performing universal computation on the DFS. Explicitly calculating the action of exchanging the physical qubits i and j on the J={fraction (3/2)}and J=½states is straightforward using (eqn). These can be written in matrix form as follows:  
           E   12     =     (         1       0       0           0         -   1         0           0       0       1         )       ,                  E   23     =     (         1       0       0           0         1   /   2             -     3       /   2             0           -     3       /   2           1   /   2           )       ,     
                       E   13     =     (         1       0       0           0         1   /   2             3     /   2             0           3     /   2             -   1     /   2           )       ,                where                 the                           
 
         [0060]    rows and columns of these matrices are labeled by the basis elements  
         {|J={fraction (3/2)},λ=0&gt;,|J=½,λ=0&gt;,|J=½,λ=1&gt;}.  
         [0061]    Therefore, the following linear combinations of these {E ij } act as the Pauli σ z  and σ x  respectively on DFS 3 (½) while annihilating the DFS 3 (3/2): {fraction (1/2)}(−E 12 +E 13 +E 23 ) and 1/{square root}{square root over (3)} (E 13 -E 23 ). However, if it is possible to perform σ z  and σ x  on a two-state system, σ y  can be performed; and these three matrices together with the identity generate all unitary gates on the system. Hence these two-body exchange operators indeed are sufficient for performing universal computation on this DFS. It is possible to show, using induction with the above argument as the base step, that two-body exchange operators are sufficient for performing universal computation on a DFS with any number of qubits larger than 2.  
         [0062]    Although the present invention has been described in detail with reference to a few embodiments, various modifications and enhancements may be made without departing from the scope and spirit of the following claims.