Patent Publication Number: US-2022222563-A1

Title: Quantum state transfer

Description:
RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Patent Application No. 63/136,385, filed Jan. 12, 2021 and titled “Fast Free-Particle Quantum State Transfer”, which is incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with government support under grant number CCF-1839232 awarded by the National Science Foundation, and grant number DESC0019040 awarded by the Department of Energy. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     The W state and Greenberger-Horne-Zeilinger (GHZ) states are two types of quantum states that entangle three qubits and can be generalized to a greater number of entangled qubits. The W state differs from the GHZ state in that if one of the qubits is lost, the remaining qubits are still entangled. On the other hand, loss of one qubit from a GHZ state causes the remaining qubits to become separable. For this reason, W states are viewed as being more robust to particle loss than GHZ states. 
     SUMMARY 
     High-fidelity quantum state transfer can be used to build fast remote quantum gates, which can significantly speed up large-scale quantum computation. There is a growing interest in designing fully-connected quantum computers that take advantage of long-range interactions among physical qubits. Finding optimal quantum state-transfer protocols using such long-range interactions is a crucial part of their design. 
     The present embodiments include a quantum state transfer protocol for spatially transferring a quantum state from a first particle of a multi-partite quantum system to a second particle of the multi-partite quantum system. The protocol works by successively spreading a particle over larger and larger regions of a lattice. The number of lattice sites supporting the particle increases exponentially with the iteration number. Specifically, after the kth iteration of the protocol at time t k , an operator c 0   †  originally supported at the origin becomes 
     
       
         
           
             
               
                 
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     After spreading the particle to a square large enough to cover both the origin and the target site, the protocol may then be reversed to compress the particle onto the target site. 
     One advantage of the quantum state transfer protocol, as compared to prior-art protocols, is that it takes place in a constrained subspace of a many-body Hilbert space that is naturally realized in atomic platforms with a conserved magnetization. Accordingly, the present embodiments can be readily implemented in many-body quantum systems having a conserved quantity. For example, in a spin system with z-spin conservation, the quantum system can be prepared in a highly polarized state with a single up-spin; the location of the up spin represents the location of the single quantum degree of freedom, and the state-transfer protocol of the present embodiments can be immediately applied. In trapped ion crystals, it is natural to use a large transverse magnetic field to help restrict to this subspace. 
     Another advantage of the present embodiments is reduced decoherence rates in the single-particle subspace. This improvement in decoherence comes from creating a series of intermediate W states. By contrast, prior-art quantum state transfer protocols utilize Greenberger-Horne-Zeilinger (GHZ) states. As an example of such a prior-art protocol, see U.S. Pat. No. 10,432,320. Also see Z. Eldredge et al., “Fast Quantum State Transfer and Entanglement Renormalization Using Long-Range Interactions”, Phys. Rev. Lett. 119, 170503 (2017). 
     Another advantage is that the protocol is extraordinarily robust to perturbations in the Hamiltonian, a desirable feature on account of the low-precision of tunable couplers used in present quantum information processors. 
     The present embodiments may be implemented with any number of physical quantum systems, including qubits and other types of quantum particles and devices. In particular, these physical implementations can be divided into at least three categories. The first category covers any many-body system in which the constituent quantum particles are, or approximate, two-level spins. In this category, XX+YY interactions between two spins transform the two-qubit state |10  into |10  (and vice versa) while leaving the states |00  and |11  unaffected. Almost all qubits used for quantum computing and information processing fall into this first category. Examples include, but are not limited to, ultracold atoms, Rydberg atoms, polar molecules, atomic ions, molecular ions, atomic nuclei, electrons, defects in diamond, and superconducting qubits (e.g., phase, charge, and flux qubits). 
     The second category of physical implementation covers any multi-partite quantum system that is, or approximates, a free boson system. Examples of free boson systems include, but are not limited to, bosonic ultracold atoms with negligible interactions and arrays of coupled cavities, such as microwave cavities (e.g., superconducting) or optical cavities. 
     The third category of physical implementation covers any multi-partite quantum system that is, or approximates, a free fermion system. Examples of free fermion systems include, but are not limited to, fermionic ultracold atoms and spin-polarized electrons. 
     In embodiments, a method for quantum state transfer includes iteratively spreading an initial quantum state from an initial qudit, of a plurality of qudits, over a sequence of expanding domains. Each iteration of said iteratively spreading includes applying a quantum circuit to the plurality of qudits belonging to a respective expanding domain of the sequence of expanding domains. The quantum circuit transforms the qudits of the respective expanding domain into an intermediate quantum state comprising a superposition of terms, at least one of the terms approximating a W state formed with the qudits of the respective expanding domain. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  illustrates a multi-partite quantum system that may be used with the present embodiments. 
         FIG. 2  illustrates a multi-spin quantum system that is an example of the multi-partite quantum system of  FIG. 1 . 
         FIG. 3  illustrates a method for quantum state transfer that can be implemented with the multi-partite quantum system of  FIG. 1 , in embodiments. 
         FIG. 4  illustrates another method for quantum state transfer that can be implemented with the multi-partite quantum system of  FIG. 1 , in embodiments. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a multi-partite quantum system  100  that may be used with the present embodiments. The quantum system  100  is formed from qudits  102  that are coupled to each other via nearest-neighbor couplings  110  and long-distance couplings  120 . Each of the couplings  110  and  120  is a direct two-body coupling, where the term “direct” means that the coupling involves only two of the qudits  102  and no intermediary qudit. The positions of the qudits  102  in  FIG. 1  approximately correspond to their positions in physical space. Thus, the qudits  102  in  FIG. 1  form a two-dimensional cubic lattice. More specifically, one qudit  102  is located at each site of the cubic lattice. However, the qudits  102  may form another structure without departing from the scope hereof. Examples of such alternative structures includes a lattice of a different dimension (e.g., one or three), a different type of lattice (e.g., triangular, hexagonal, rectangular, etc.), a structure with irregular spacings between the qudits  102 , and combinations thereof. 
     Each of the nearest-neighbor couplings  110  couples one of the qudits  102  to one of the qudits  102  to which it is physically closest. For example, in  FIG. 1  a first nearest-neighbor coupling  110 ( 1 ) couples a first qudit  102 ( 1 ) with a second qudit  102 ( 2 ). Similarly, a second nearest-neighbor coupling  110 ( 2 ) couples the first qudit  102 ( 1 ) with a third qudit  102 ( 3 ). The qudits  102 ( 2 ) and  102 ( 3 ) are the two qudits physically closest to the first qudit  102 ( 1 ), and are therefore nearest-neighbor qudits with respect to the first qudit  102 ( 2 ). As shown by this example, it is possible for a qudit to have more than one nearest-neighbor qudit. For clarity in  FIG. 1 , only the nearest-neighbor couplings  110 ( 1 ) and  110 ( 2 ) are labeled. However, the dashed lines in  FIG. 1  indicate many other nearest-neighbor couplings. 
     Each of the long-range couplings  120  couples one of the qudits  102  to another qudit that is not a nearest-neighbor qudit. For example, in  FIG. 1  a first nearest-neighbor coupling  120 ( 1 ) couples the first qudit  102 ( 1 ) with a fourth qudit  102 ( 4 ) and a second nearest-neighbor coupling  120 ( 2 ) couples the first qudit  102 ( 1 ) with a fifth qudit  102 ( 5 ). As can be seen, the qudits  102 ( 4 ) and  102 ( 5 ) are not nearest-neighbor qudits with respect to the first qudit  102 ( 1 ). 
     Each of the couplings  110  and  120  can be represented as a term in the Hamiltonian of the quantum system  100 . The term may include a coupling strength that quantifies its contribution to the Hamiltonian. While  FIG. 1  only shows direct two-body couplings, those skilled in the art will recognize that the quantum system  100  may alternatively or additionally include direct higher-order couplings (e.g., three-body couplings that directly involve three of the qudits  102 , four-body couplings that directly involve four of the qudits  102 , etc.). 
     The quantum system  100  also includes indirect couplings between the qudits  102 . Here, the term “indirect” means that the coupling involves intermediary qudits. Thus, two qudits  102  that are not directly coupled are still indirectly coupled through a chain of two more direct couplings. Therefore, each qudit  102  in the quantum system  100  is coupled, either directly or indirectly, to every other qudit  102  in the quantum system  100 . It is not necessary that the quantum system  100  include both nearest-neighbor couplings  110  and long-range couplings  120 . For example, the quantum system  100  may have only nearest-neighbor couplings  110  and no long-range couplings  120 . In another example, the quantum system  100  only has long-range couplings  120  and no nearest-neighbor couplings  110 . 
     Each of the qudits  102  is a quantum particle that, in the absence of the couplings  110  and  120 , has at least d discrete internal energy levels that can be used to represent quantum information, where d is an integer greater than or equal to 2. When d=2, each of the qudits  102  is, or approximates, a qubit. When d=3, each of the qudits  102  is, or approximates, a qutrit. More generally, the present embodiments can be implemented with qudits  102  having any value of d≥2. Furthermore, each of the qudits  102  may have more internal energy levels than necessary to represent quantum information (e.g., a quantum particle with d=3 can be used as a qubit). Examples of the qudits  102  include, but are not limited to, ultracold atoms, Rydberg atoms, trapped ions, electrons, atomic nuclei, superconducting circuits (e.g., phase, charge, and flux qubits), defects in diamond (e.g., nitrogen vacancies), photons, electromagnetic cavities (e.g., optical or microwave), or a combination thereof. 
     The qudits  102  may be any combination of bosons and fermions. In some embodiments, all of the qudits  102  are bosons. In one of these embodiments, the quantum system  100  is, or approximates, a free boson system that contains only two-body couplings and no higher-order couplings. In this case, the Hamiltonian of the quantum system  100  can be diagonalized to identify quasiparticles. In other embodiments, all of the qudits  102  are fermions. In one of these embodiments, the quantum system  100  is, or approximates, a free fermion system. 
       FIG. 2  illustrates a multi-spin quantum system  200  formed from interacting spins  202 . The quantum system  200  is an example of the multi-partite quantum system  100  of  FIG. 1 . Each of the spins  202  is a qubit, and therefore an example of a qudit  102  for the case of d=2. The spins  202  interact with each other via spatial overlap of their wavefunctions, giving rise to nearest-neighbor couplings and long-range couplings like those shown in  FIG. 1 . Generally, the strength of these interactions, and hence the coupling strength, decreases with increasing distance. Thus, the spins  202  exhibit inherent coupling, i.e., coupling that arises simply by placing the spins  202  proximate to each other. Each spin  202  may be a quantum particle having a magnetic dipole moment (e.g., an electron), electric dipole moment (e.g., Rydberg atom), or any two internal energy levels whose dynamics can be described by quantum spin. 
     As a contrast to the spins  202 , each of the qudits  102  may be a quantum particle or device without inherent coupling. In this case, the qudits  102  may be placed proximate to each other with negligible interactions therebetween. The quantum system  100  may include couplers that provide coupling between qudits  102 . For some types of the qudits  102 , such couplers may be controlled (via an electrical signal) to selectively enable and disable coupling between specific pairs of qudits  102 . For example, many superconducting qubits have negligible inherent coupling, and circuits using superconducting qubits include elements that introduce coupling. Such elements may be as a simple as a passive capacitor or transmission line. One such element that is controllable is a Josephson junction. When the Josephson junction is activated, it generates a magnetic field that shifts the energies of the qubits&#39; internal levels out of resonance with the energy levels of a neighboring qubit, thereby making coupling therebetween negligible. Other examples of quantum particles without inherent coupling include electromagnetic cavities. For example, microwave cavities can be coupled to each other using transmission lines or waveguides. As another example, optical cavities can be coupled to each other using optical fibers or free-space coupling. 
       FIG. 3  illustrates a method  300  for quantum state transfer that can be implemented with the quantum system  100  of  FIG. 1 . The method  300  iteratively spreads an initial quantum state from an initial qudit  302  over a sequence of expanding domains. For clarity, it is assumed in the following discussion that each of the qudits  102  is a qubit with computational basis states |0  and |1 . However, those trained in the art should recognize how to implement the method  300  with other types of qudits. Also for clarity, the couplings  110  and  120  between qudits  102  are not shown. 
     In  FIG. 3 , the qudits  102  are grouped into an expanding sequence B (e)  of N domains. Mathematically, this expanding sequence B (e)  can be denoted: 
         B   (e) =( B   i   (e) ) i=1   N =( B   1   (e)   , B   2   (e)   , . . . , B   N−1   (e)   , B   N   (e) )  (1)
 
     where B i   (e)  denotes the i th  domain, i is an index that runs from 1 to N, and the superscript “e” indicates expansion. Each domain B i   (e)  is a connected subset of the qudits  102 , where “connected” means that each qudit  102  in the domain B i   (e)  is coupled, either directly or indirectly, to every other qudit  102  in the domain B i   (e) . The domains are “expanding” in that each one is a superset of all preceding domains of the sequence B (e)  and a subset of all subsequent domains of the sequence B (e) . Thus, the expanding domains satisfy 
         B   1   (e)   ⊂B   2   (e)   ⊂ . . . ⊂B   N−1   (e)   ⊂B   N   (e) .  (2)
 
     While Eqn. 2 shows each domain B i   (e)  as a proper subset of all subsequent domains, one or more of the domains may be equal to one or more subsequent domains provided that the first domain B 1   (e)  is a proper subset of the final domain B N   (e) . 
     In  FIG. 3 , the first domain B 1   (e)  of the expanding sequence B (e)  includes only an initial qudit  302 . The second domain B 2   (e)  has four qudits  102 , one of which is the initial qudit  302 . Similarly, the third domain B 3   (e)  has nine qudits  102 , including all four qudits  102  belonging to the second domain B 2   (e) . Finally, the fourth domain B 4   (e)  has twenty-five qudits  102 , including all nine qudits  102  belonging to the third domain B 3   (e) . While N=4 for the example of  FIG. 3 , a different value of N may be used without departing from the scope hereof. The value of N may be chosen based on the overall number of qudits  102 , distances between qudits  102 , couplings between qudit  102 , type of qudits  102 , and other parameters. 
     The method  300  includes iteratively applying a sequence U (e)  of N−1 quantum circuits to the qudits  102 . The sequence U (e)  can be represented mathematically as 
         U   (e) =( U   i   (e) ) i=1   N−1 =( U   1   (e)   , U   2   (e)   , . . . , U   N−1   (e) ).  (3)
 
     In each iteration, a quantum circuit U i   (e)  is applied to one or more of the qudits  102  belonging the domain B i+1   (e) . Each quantum circuit implements any combination of quantum logic gates, initialization, coupling, and measurements. In some embodiments, each quantum circuit implements a unitary transformation. Since a unitary transformation is reversible, the effect of the quantum circuit is deterministic. In other embodiments, each quantum circuit is probabilistic (e.g., being based on the outcome of a measurement performed with the qudits  102 ). 
     At the beginning of the method  300 , the initial qudit  302  is in an initial state |Ψ 0   =α|0 +β|1 , where α and β are complex coefficients satisfying |α| 2 +|β| 2 =1. All of the other qudits  102  have been initialized into the |0  state. In the first iteration of the sequence U (e) , the first quantum circuit U 1   (e)  is applied to the qudits  102  belonging to the second domain B 2   (e)  to transform these qudits  102  into a first expanded state |Ψ 1    that is a superposition of a W state and a multi-partite all-zero state. Mathematically, this transformation can be expressed as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     where |W 1     =(| 1000 +|0100 +|0010 +|0001 )/2 is a first W state formed from the four qudits  102  belonging to the second domain B 2   (e)  and |Z 1   =|0000  is a first all-zero state that is also formed from the four qudits  102  belonging to the second domain B 2   (e) . The first W state is maximally entangled, meaning that its constituent terms have equal coefficients. When α=0 and β=1, the first quantum circuit U 1   (e)  transforms the initial state |Ψ 0   =|1  into the W state |W 1   . When α=1 and β=0, the first quantum circuit U 1   (e)  transforms the initial state |Ψ 0     =|0    into the all-zero state |Z 1   . More generally, a maximally entangled W state |W  formed from n qubits can be represented mathematically as 
     
       
         
           
             
               
                 
                   
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     while an all-zero state |Z  formed from n qubits can be represented mathematically as 
       | Z     =| 0   ⊗n =|0 ⊗|0 ⊗ . . . ⊗|0 .  (6)
 
     In the second iteration of the sequence U (e) , the second quantum circuit U 2   (e)  is applied to the qudits  102  belonging to the third domain B 3   (e)  to transform these qudits  102  into a second expanding state |Ψ 2    that is also a superposition of a W state and an all-zero state. Mathematically, 
       |Ψ 2     =U   2   (e) ((α| Z   1   +β|W 1   )⊗|00000 )=α| z   2     +β|W   2   .  (7)
 
     In Eqn. 7, |W 2 ) is a second W state formed from the nine qudits  102  belonging to the domain B 3   (e)  and |Z 2   =|000000000  is a second all-zero state that is also formed from the nine qudits  102  belonging to the domain B 3   (e) . 
     In the i th  iteration of the sequence U (e) , the i th  quantum circuit U i   (e)  is applied to the qudits  102  belonging to the domain B i+1   (e)  to transform these qudits  102  into an i th  expanded state |Ψ i    that is also a superposition of an i th  W state and an i th  all-zero state, or 
       |Ψ i     =U   i   (e) ((α| W   i−1     +β|Z   i−1   )⊗| Z   i′ ))=α| W   i     +β|Z   i   .  (8)
 
     In Eqn. 8, |Z i′ ) is an all-zero state of the qudits  102  that are new to the domain B i+1   (e) , i.e., the qudits  102  that belong to the domain B i+1   (e)  and do not belong to the preceding domain B i   (e) . Thus, the state |Z i′ ) is the product of the initial state |0  of each of the new qubits. As can be seen from Eqn. 8, the input to the quantum circuit U i   (e)  is the product of (i) the intermediate quantum state |Ψ i   =α|W i−1   +β|Z i−1    generated by the preceding quantum circuit U i−1   (e)  and (ii) the all-zero state |Z i′   . 
     In the last iteration of the sequence U (e) , the final quantum operator U N−1   (e)  is applied to the qudits  102  belonging to the final domain B N   (e)  to obtain a maximally expanded state |Ψ N−1   =α|W N−1   +β|Z N−1    that is a superposition of a maximum W state |W N−1    and a maximum all-zero state |W N−1   . The state |Ψ N−1    is “maximally expanded” in that it involves more entangled qudits  102  than any of the other expanded states |Ψ 1   , |Ψ 2   , . . . , |Ψ N−2   . 
     When the qudits  102  do not possess intrinsic coupling, each quantum circuit may control one or more couplers to couple the qudits  102  (i.e., “turn on” the couplings), adjust coupling strengths, and decouple the qudits  102  (i.e., “turn off” the couplings). In some embodiments of the method  300 , the quantum circuit U i   (e)  generates a plurality of pair-wise couplings between (i) one or more of the qudits  102  that belong to the previous domain B i   (e)  and (ii) one or more of the qudits  102  that belong to the domain B i+1   (e)  and are excluded from the previous domain B i   (e) . The plurality of pair-wise couplings may be generated simultaneously. The pair-wise couplings may fully connect the (i) one or more of the qudits  102  that belong to the previous domain B i   (e)  and the (ii) one or more of the qudits  102  that belong to the domain B i+1   (e)  and are excluded from the previous domain B i   (e) . In other embodiments, the qudits  102  possess intrinsic coupling. In these embodiments, the method  300  may exclude the turning on and off of pair-wise couplings. 
     In some embodiments, the plurality of qudits  102  form a lattice with dimension d. The number of the qudits belonging to the domain B i   (e)  is approximately 2 d  times the number of the qudits belonging to the previous domain B i−1   (e) . For example, in  FIG. 3 , where d=2, the four qudits in B 2   (e)  is exactly four times the number of qudits in B 1   (e) , the nine qudits in B 3   (e)  is approximately four times the number of qudits in B 2   (e) , and the twenty-five qudits in B 4   (e)  is approximately four times the number of qudits in B 3   (e) . As can be seen from this example, this increase in the number of qudits by a factor of 2 d  from each domain to the next need not be exact. 
     In other embodiments, the method  300  further includes assigning each of the qudits  102  to one or more of the expanding domains B i   (e) . In some embodiments, the method  300  includes initializing, prior to said iteratively spreading, the initial qudit  302  to have the initial quantum state |Ψ 0   . In one of these embodiments, the method  300  includes initializing, prior to said iteratively spreading, all of the qudits  102 , except the initial qudit  302 , into a local eigenstate. For example, when the qudits  102  are qubits, the local eigenstate may be the computational basis state |0 . 
     In some embodiments, the method  300  further includes initializing the initial qudit  302  into the initial state |Ψ 0   =|1 . In this case, the maximally expanded state |Ψ N−1    is approximately the maximum W state |W N−1    (i.e., the maximum all-zero state |Z N−1    is negligible and therefore can be ignored). The method  300  may then further include generating a single photon with the maximum W state |W N−1   . Compared to other types of entangled states, W states can be converted more easily into single photons because they couple to light with a coupling constant that can be collectively enhanced (e.g., by the square-root of the number of qudits  102 ). In other embodiments, the method  300  further includes receiving a single photon with the qudits  102  forming the maximum W state |W N−1   . Thus, these embodiments can be used to create a matter-photon interface. Such an interface may form part of a quantum repeater or quantum network. 
     In another application, the method  300  is used to convert between W states and localized excitations to process photonic states. An example of how such a conversion may be implemented can be found in Christine A. Muschik et al., “Quantum Processing Photonic States in Optical Lattices”, Phys Rev. Lett. 100, 063601. 
       FIG. 4  illustrates a method  400  for quantum state transfer that can be implemented with the quantum system  100  of  FIG. 1 . The method  400  extends the method  300  of  FIG. 3  by iteratively compressing the maximally expanded state |Ψ N−1    over a sequence of contracting domains. The method  400  can be used to transfer the initial quantum state |Ψ 0    from the initial qudit  302  to a final qudit  402 . Similar to the above discussion for  FIG. 3 , it is assumed in the following discussion that each of the qudits  102  is a qubit with computational basis states |0  and |1 . Those trained in the art should recognize how to implement the method  400  with other types of qudits. Similarly, the couplings  110  and  120  between the qudits  102  are not shown in  FIG. 4 . 
     In  FIG. 4 , the qudits  102  are grouped into a contracting sequence B (c)  of M domains. Mathematically, the contracting sequence B (c)  can be expressed as 
         B   (c) =( B   i   (c) ) i=1   M =( B   1   (c)   , B   2   (c)   , . . . , B   M−1   (c)   , B   M   (c) ),  (9)
 
     where B i   (c)  denotes the i th  domain, i is an index that runs from 1 to M, and the superscript “c” indicates contraction. Each domain B i   (c)  is a connected subset of the qudits  102 . The domains are “contracting” in that each one is a subset of all preceding domains of the sequence B (c)  and a subset of all subsequent domains of the sequence B (c) . Thus, the contacting domains satisfy 
       B 1   (c) ⊃B 2   (c) ⊃ . . . ⊃B M−1   (c) ∈B M   (c) .  (10)
 
     While Eqn. 10 shows each domain B i   (c)  as a proper superset of all subsequent domains, one or more of the domains may be equal to one or more subsequent domains provided that the first domain B 1   (c)  is a proper superset of the final domain B N   (c) . 
     In  FIG. 4 , the first domain B 1   (c)  of the contracting sequence B (c)  includes all of the qudits  102  of the final domain B N   (e)  of the expanding sequence B (e) . The domain B 1   (c)  is the only domain of the sequence B (c)  that contains the initial qudit  302 . The second domain B 2   (c)  has nine qudits  102 , the third domain B 3   (c)  has four qudits  102 , and the fourth domain B 4   (c)  contains only the final qudit  402 . While M=4 for the example of  FIG. 4 , a different value of M may be used without departing from the scope hereof. The value of M may be chosen based on the overall number of qudits  102 , distances between qudits  102 , couplings between qudit  102 , type of qudits  102 , and other parameters. It is possible, although not necessary, for M to equal to the same value of N used for the method  300 . 
     In the method  400 , the method  300  is first used to generate the maximally expanded state |Ψ N−1   . A sequence U (c)  of M−1 quantum circuits is then applied iteratively to the qudits  102 . The sequence U (c)  can be represented mathematically as 
         U   (c) =( U   i   (c) ) i=1   M−1 =( U   1   (c)   , U   2   (c)   , . . . , U   M−1   (c) ).  (11)
 
     In each iteration, a quantum circuit U i   (c)  is applied to one or more of the qudits  102  belonging the domain B i   (c) . When the qudits  102  do not possess intrinsic coupling, the action of a quantum circuit may include controlling one or more couplers to couple the qudits  102 , adjust coupling strengths, and decouple the qudits  102 . When the qudits  102  possess intrinsic coupling, the action of a quantum circuit may exclude the turning on and off of couplings. 
     In the first iteration of the sequence U (c) , the first quantum circuit U 1   (c)  is applied to the twenty-five qudits  102  belonging to the first domain B 1   (c)  to transform these qudits  102  from the maximally expanded state |Ψ N−1    into a first contracted state |Ψ′ 1    that is a superposition of a W state and a multi-partite all-zero state. Mathematically, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           
                             ψ 
                             1 
                             ′ 
                           
                           〉 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             U 
                             1 
                             
                               ( 
                               c 
                               ) 
                             
                           
                           ⁢ 
                           
                              
                             
                               ψ 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             〉 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             U 
                             1 
                             
                               ( 
                               c 
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 α 
                                 ⁢ 
                                 
                                    
                                   
                                     Z 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                   〉 
                                 
                               
                               + 
                               
                                 β 
                                 ⁢ 
                                 
                                    
                                   
                                     W 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                   〉 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               U 
                               1 
                               
                                 ( 
                                 c 
                                 ) 
                               
                             
                             ⁢ 
                             
                                
                               
                                 Z 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               〉 
                             
                           
                           + 
                           
                             β 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               U 
                               1 
                               
                                 ( 
                                 c 
                                 ) 
                               
                             
                             ⁢ 
                             
                                
                               
                                 W 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               〉 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 α 
                                 ⁢ 
                                 
                                    
                                   
                                     Z 
                                     1 
                                     ′ 
                                   
                                   〉 
                                 
                               
                               + 
                               
                                 β 
                                 ⁢ 
                                 
                                    
                                   
                                     W 
                                     1 
                                     ′ 
                                   
                                   〉 
                                 
                               
                             
                             ) 
                           
                           ⊗ 
                           
                              
                             
                               Z 
                               1 
                               * 
                             
                             〉 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where |W′ 1    is a first W state formed from the qudits  102  belonging to the second domain B 2   (c)  and |Z′ 1 ) is a first all-zero state formed from the qudits  102  belonging to the second domain B 2   (c) . The qudits  102  that belong to the first domain B 1   (c)  but are excluded from the second domain B 2   (c)  are called separated qudits  102 , and the quantum circuit U 1   (c)  decouples these separated qudits  102  such that they are no longer entangled with the other qudits  102 . Each separated qudit  102  is in the state |0 , and collectively these separated qudits are indicated in Eqn. 12 with the all-zero state |Z* 1   . The separated qudits  102  are no longer needed, and therefore may be ignored or repurposed. 
     In the second iteration of the sequence U (c) , the second quantum circuit U 2   (c)  is applied to the nine qudits  102  belonging to the second domain B 2   (c)  to transform these qudits  102  into a second contracting state |Ψ′ 2    that is also a superposition of a W state and an all-zero state. Ignoring the separated qudits in Eqn. 12, the second quantum circuit U 2   (c)  is applied to only those qudits  102  in the state α|Z′ 1   +β|W′ 1   . Mathematically, 
       |Ψ′ 2     =U   2   (c) (α| Z′   1     +β|W′   1   )=(α| Z′   2     +β|W′   2   )⊗| Z*   2     (13)
 
     where |W′ 2    is a second W state formed from the four qudits  102  belonging to the third domain B 3   (c) , |Z′ 2   =|0000  is a second all-zero state formed from the four qudits  102  belonging to the third domain B 3   (c) , and |Z* 2    represents qudits  102  in the domain B 2   (c)  that have been separated. Again, the separated qudits are no longer needed and therefore can be ignored or repurposed. 
     In the i th  iteration of the sequence U (c) , the i th  quantum circuit U i   (c)  is applied to the qudits  102  belonging to the domain B i   (c)  to transform these qudits  102  into an i th  contracting state |Ψ′ i   . Mathematically, 
       |Ψ′ i     =U   i   (c) (α| Z′   i−1     +β|W′   i−1   )=(α| Z′   i     +β|W′   i   )⊗| z*   i     (13)
 
     where |W i   t    is an i th  W state formed from the qudits  102  belonging to the domain B i   (c) , |Z′ i    is an i th  all-zero state formed from the qudits  102  belonging to the domain B i   (c) , and |Z* i    represents qudits  102  in the domain B i   (c)  that have been separated. 
     In the last iteration of the sequence U (c) , the final quantum operator U M−1   (c)  is applied to the qudits  102  belonging to the final domain B M−1   (c) . After ignoring additional separated qudits, the final qudit  402  is in a final quantum state |Ψ f    that approximates the initial quantum state |Ψ 0   , i.e., |Ψ f   ≈|Ψ 0   . Thus, the method  400  spatially transfers the initial quantum state |Ψ 0    across the distance between the initial qudit  302  and the final qudit  402 . 
     In  FIG. 4 , the method  400  compresses the same qudits  102  that were used for spreading in  FIG. 3 . However, spreading and compression may use at least partially different sets of qudits  102 . Accordingly, in some embodiments the method  400  includes assigning each of a first plurality of qudits to one or more of the expanding domains, and assigning each of a second plurality of qudits to one or more of the contracting domains. Both the initial qudit  302  and final qudit  402  belong to both the first and second pluralities of qudits. Accordingly, there are at least two qudits common to both the first and second pluralities of qudits. In one embodiment, the second plurality of qudits is the same as the first plurality of qudits. 
     In  FIG. 3 , the initial qudit  302  lies on a boundary each of the domains B i   (e) , including the largest domain B N−1   (e) . Similarly, in  FIG. 4 , the final qudit  402  lies on a boundary of each of the domains B i   (c) , including the largest domain B 1   (c)  However, it is not necessary for the initial qudit  302  to lie on any boundary, except for the first domain B 1   (e)  when the initial qudit  302  is the only qudit belonging thereto. Similarly, it is not necessary for the final qudit  402  to lie on any boundary, except for the last domain B M   (c)  when the final qudit  402  is the only qudit belonging thereto. Accordingly, in some embodiments of the methods  300  and  400 , one or both of the qudits  302  and  402  do not lie on the boundary of the largest domain. In another embodiment, the initial qudit  302  lies on the boundary of only the first domain B 1   (e)  and the final qudit  402  lies on the boundary of only the last domain B M   (c) . 
     In some embodiments, the plurality of qudits  102  form a lattice with dimension d. The number of the qudits belonging to the domain B i   (c)  is approximately 2 −d  times the number of the qudits belonging to the previous domain B i−1   (c) . For example, in  FIG. 4 , where d=2, the twenty-five qudits in B 1   (c)  is approximately four times the number of qudits in B 2   (c) , the nine qudits in B 2   (c)  is approximately four times the number of qudits in B 3   (c) , and the four qudits in B 3   (c)  is exactly four times the number of qudits in B 4   (c) . As can be seen from this example, this reduction of the number of qudits by the factor 2 −d  from each domain to the next need not be exact. 
     To better understand the quantum circuits used in the present embodiments, consider a many-body quantum system defined on a d-dimensional lattice graph Λ of qudits (such as the quantum system  100  of  FIG. 1  or the quantum system  200  of  FIG. 2 ). The Hamiltonian of this quantum system may take the form 
     
       
         
           
             
               
                 
                   
                     H 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         , 
                         
                           j 
                           ∈ 
                           Λ 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           h 
                           ij 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                         c 
                         i 
                         † 
                       
                       ⁢ 
                       
                         c 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     where h(t): →   Λ×Λ  is a Hermitian matrix, and c t   †  and c i  represent either the fermionic creation and annihilation operators obeying 
       { c   j   c   i   † }=δ ij   (16)
 
     or the bosonic creation and annihilation operators obeying 
       [ c   j   ,c   i   † ]=δ ij .  (17)
 
     The time-dependence of the matrix h(t) reflects the turning on and off of pair-wise couplings between pairs of the qudits. The diagonal elements h ij  do not participate in spreading and compression and therefore may be ignored. The on-site Hilbert space    i  obeys dim(   i )=2 in the fermionic case and dim(   i )=∞ in the bosonic case. However, in isolated bosonic systems    i  can often be truncated so that dim(   i ) is at most the number of excitations on the lattice and is therefore finite. 
     As known in the art, the evolution of all operators in such a non-interacting theory is controlled by the Green&#39;s function of the single particle problem on the Hilbert space    Λ . Time evolution on this space is generated by the Hamiltonian 
     
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         sp 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           
                             j 
                             ∈ 
                             Λ 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             h 
                             ij 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                            
                           
                             1 
                             i 
                           
                           〉 
                         
                         ⁢ 
                         
                           〈 
                           
                             1 
                             j 
                           
                            
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where |1 i    denotes the many-body quantum state having one excitation at site i ∈ Λ and no excitations at all other sites. The single particle time evolution matrix obeys the differential equation 
     
       
         
           
             
               
                 
                   
                     
                       
                         d 
                         dt 
                       
                       ⁢ 
                       
                         
                           U 
                           sp 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             iH 
                             sp 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           U 
                           sp 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     together with the initial condition U sp (0)=1. For example, in the fermionic model, 
     
       
         
           
             
               
                 
                   
                     
                       
                         c 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           j 
                           ∈ 
                           Λ 
                         
                       
                       ⁢ 
                       
                         
                           
                             U 
                             
                               sp 
                               , 
                               ij 
                             
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                           c 
                           j 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     from which it follows that 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       dt 
                     
                     ⁢ 
                     
                       c 
                       i 
                     
                   
                   = 
                   
                     
                       i 
                       ⁡ 
                       
                         [ 
                         
                           
                             H 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             c 
                             i 
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             Λ 
                           
                         
                         ⁢ 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 h 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     c 
                                     i 
                                     † 
                                   
                                   ⁢ 
                                   
                                     c 
                                     j 
                                   
                                 
                                 , 
                                 
                                   c 
                                   i 
                                 
                               
                               ] 
                             
                           
                         
                       
                       = 
                       
                         i 
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               Λ 
                             
                           
                           ⁢ 
                           
                             
                               
                                 h 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 c 
                                 j 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Next, let A and B be two disjoint subsets of A and 0&lt;C&lt;∞. If 
     
       
         
           
             
               
                 
                   
                      
                     
                       ψ 
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     〉 
                   
                   = 
                   
                     
                       1 
                       
                         
                            
                           A 
                            
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           A 
                         
                       
                       ⁢ 
                       
                          
                         
                           1 
                           i 
                         
                         〉 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     then there exists a free-particle Hamiltonian H(t) defined in Eqn. 15 with |h ij |≤C for all i, j ∈ Λ such that for any θ ∈   
     
       
         
           
             
               
                 
                   
                      
                     
                       ψ 
                       ⁡ 
                       
                         ( 
                         T 
                         ) 
                       
                     
                     〉 
                   
                   ∝ 
                   
                     
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         
                           
                              
                             A 
                              
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             A 
                           
                         
                         ⁢ 
                         
                            
                           
                             1 
                             i 
                           
                           〉 
                         
                       
                     
                     + 
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         
                           
                              
                             B 
                              
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             B 
                           
                         
                         ⁢ 
                         
                            
                           
                             1 
                             i 
                           
                           〉 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     at time 
     
       
         
           
             
               
                 
                   T 
                   ≤ 
                   
                     
                       π 
                       
                         2 
                         ⁢ 
                         C 
                         ⁢ 
                         
                           
                             
                                
                               B 
                                
                             
                             ⁢ 
                             
                                
                               A 
                                
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     This assertion can be proven by construction. Consider the Hamiltonian 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       e 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       sgn 
                       ⁡ 
                       
                         ( 
                         
                           tan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           A 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             B 
                           
                         
                         ⁢ 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               C 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                        
                                       
                                         1 
                                         j 
                                       
                                       〉 
                                     
                                     ⁢ 
                                     
                                       〈 
                                       
                                         1 
                                         k 
                                       
                                        
                                     
                                   
                                   - 
                                   
                                     
                                        
                                       
                                         1 
                                         k 
                                       
                                       〉 
                                     
                                     ⁢ 
                                     
                                       〈 
                                       
                                         1 
                                         j 
                                       
                                        
                                     
                                   
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Without loss of generality, we take θ ∈[0, π/2]; the generalization to other values of θ is straightforward. By permutation symmetry, the wave function takes the form of Eqn. 23 with θ (t) a function of time. Pick any j ∈ B. We can explicitly evaluate 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     dt 
                   
                   = 
                   
                     
                       
                         
                           
                              
                             B 
                              
                           
                         
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       ⁢ 
                       
                         
                           
                             d 
                             ⁢ 
                             
                               〈 
                               j 
                                
                             
                             ⁢ 
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           〉 
                         
                         dt 
                       
                     
                     = 
                     
                       
                         
                           - 
                           i 
                         
                         ⁢ 
                         
                           
                             
                                
                               B 
                                
                             
                           
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         ⁢ 
                         
                           〈 
                           
                             j 
                             ⁢ 
                             
                                
                               
                                 H 
                                 e 
                               
                                
                             
                             ⁢ 
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           〉 
                         
                       
                       = 
                       
                         C 
                         ⁢ 
                         
                           
                             
                               
                                  
                                 B 
                                  
                               
                               ⁢ 
                               
                                  
                                 A 
                                  
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Since the value of θ at which |Ψ(t)  is given by Eqn. 23 is in the range [0, π/2], we conclude that Eqn. 26 implies Eqn. 24. 
     The state |Ψ(0)  of Eqn. 22 is a maximally entangled W state formed from the qudits in the set A. For θ=π/4, the state |Ψ(T)  of Eqn. 23 gives a maximally entangled W state formed from the qubits in the set A+B. For θ=π/2, |Ψ(T)  is a maximally entangled W state formed from the qubits in the set B For other values of θ, |Ψ(T)  is a partially entangled W state formed from the qubits in the set A+B. 
     The Hamiltonian H e  of Eqn. 23 is an example of the Hamiltonian H(t) of Eqn. 15, i.e., a Hamiltonian that transforms the state |Ψ(0)  of Eqn. 22 into the state |Ψ(T)  of Eqn. 23. Those trained in the art will recognize that other Hamiltonians can produce similar transformations. In particular, the Hamiltonian H e  fully couples the qubits of the set A with the qubits of the set B, where “fully couples” means that each of the qubits in the set A is directly two-body coupled to each of the qubits in the set B. Fully coupled qubits generally gives the fastest transformation times. However, the present embodiments do not require the qubits to be fully coupled (i.e., some of the qubits may be indirectly coupled, as described above). 
     The Hamiltonian H(t) has no effect on a multi-zero state. Therefore, for each iteration i of iteratively spreading, the respective quantum circuit U i   (e)  implements a Hamiltonian of the form shown in Eqn. 15. Furthermore, when |Ψ(0)  is a maximally entangled W state over the qubits of A+B, the Hamiltonian H e  will transform this state into a maximally entangled W state over just the qubits of B. Therefore, any Hamiltonian H(t) of Eqn. 15 used for spreading can also be used for compression. 
     Those trained in the art will recognize how to experimentally implement a Hamiltonian of the form shown in Eqn. 15. The details of any implementation depend on the type of qudits used. For example, when the qudits are superconducting circuits, a pair of such superconducting circuits may be coupled together using a capacitor, transmission line, or cavity bus. In this case, one or both of superconducting circuits may be detuned to effectively turn off the coupling. Alternatively, a DC SQUID or Josephson junction (e.g., flux biased or current biased) may be used to controllably couple the qudits. In any case, a coupler between the i th  qudit and j th  qudit of the quantum system  100  can be described mathematically as the term h ij  of the Hamiltonian H(t) of Eqn. 15. Accordingly, the full Hamiltonian H(t) can be experimentally realized by coupling pairs of the qudits  102  and controlling these couplings over time. 
     As another example, the qudits  102  may be a chain of trapped ions. Two internal energy states of each ion serve as the computational basis states |0  and |1 . A laser pulse couples a first ion to a collective motional mode of the chain. The same laser pulse also couples the collective motional mode to a second ion. This laser-driven process, referred to as “red-sideband coupling” and “blue-sideband coupling”, therefore couples the first and second ions using the collective motional mode as an intermediary. Since these couplings only occur in the presence of laser light, they are inherently controllable. Furthermore, such coupling between a pair of ions can be expressed mathematically as one of the terms h ij  in the Hamiltonian H(t) of Eqn. 15 (or, equivalently, one of the terms in the Hamiltonian H e  of Eqn. 25). 
     For any one of the quantum circuits U i   (e)  and U i   (c) , the couplings may be applied simultaneously, sequentially, or a combination thereof. Applying the coupling simultaneously usually results in the fastest transfer times. However, in certain physical implementations it may not be possible to simultaneously apply all couplings. In this case, the couplings may be applied sequentially provided that the overall effect of the quantum circuit is unaffected. 
     One benefit of the method  400  is its remarkable robustness to error. As a heuristic argument for this robustness, consider the case where the number   of qudits  102  in the domain B i+1   (e)  is approximately  =2 di . Also assume that these   qudits are mutually coupled. The method benefits from this mutually coupling via an enhancement in the transfer rate of a factor of  . Consider an uncorrelated random error in h(t). Using random matrix theory, such an error leads to dephasing rates of order  . Denoting |x  as the target site (i.e., the final qudit  402 ), the loss in fidelity  =| Ψ(τ)|x | 2  can be estimated by summing up the error after each step: 
     
       
         
           
             
               
                 
                   
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                   27 
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     Here, ϵ is related to the error in a single coupling. Therefore, the spreading and compression performed by the method  400  is highly immune to imperfections in tunable coupling constants. As ϵ→0, the fidelity  →1. 
     The present embodiments also include systems that perform the methods  300  and  400 . In one such embodiment, a system for quantum state transfer includes a plurality of qudits and a plurality of couplers that couple the qudits. The qudits may take the form of a quantum register. Each of the couplers may be controllable to directly couple two of the qudits. The system may also include one or more instruments that control any combination of the qudits and couplers to establish coupling between the qudits in accordance with one or both of the methods  300  and  400 . These instruments may be specific to the type of qudits employed in the system. For example, when the qudits are superconducting circuits, the instruments may include microwave signal generators or synthesizers that output signals to be applied to the qudits and tunable couplers. When the qudits are trapped atoms or ions, the instruments may include lasers and associated optical components (e.g., modulators) for generating a light beam steering light beam to the trapped atoms or ions. Other types of instruments include, but are not limited to, DC current sources (e.g., for generating magnetic fields with coils) and DC voltage sources (e.g., for driving electrodes). The system may include other instruments without departing from the scope hereof. 
     The system may also include a computer or controller that controls the instruments according to the quantum circuits described herein. The computer may include a processor and a memory storing machine-readable instructions that, when executed by the processor, control the instruments such that the system performs one or both of the methods  300  and  400 . The processor may include one or more of a microprocessor with one or more central processing unit (CPU) cores, a graphics processing unit (GPU), a digital signal processor (DSP), a system-on-chip (SoC), a microcontroller unit (MCU), or another type of circuit that executes instructions to perform logic, control, and input/output operations. The controller may include a field-programmable gate array (FPGA) or other type of hard-wired circuit that is pre-programmed to control the instruments such that the system performs one or both of the methods  300  and  400 . The controller may be part of a data acquisition system that processes signals (e.g., measurements of the qudits). 
     As an example of its use, the system for quantum state transfer may form part of a quantum computer or information processing system. As another example, the system may form part of a device used for quantum communication. Examples of such devices include quantum repeaters, quantum memories, quantum transmitters, and quantum repeaters. More generally, the system may be used in any quantum device or experimental setup in which a quantum state is transferred from one qubit to another. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.