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SIAM Journal on Computing

Stabilization of Quantum Computations by Symmetrization

Adriano Barenco, André Berthiaume, David Deutsch, ARTUR K. EKERT, Richard Jozsa, Chiara Macchiavello

Published Oct 1, 1997• 238 citations
Publisher

Abstract

We propose a method for the stabilization of quantum computations (including quantum state storage). The method is based on the operation of projection into $\cal SYM$, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting $\cal SYM$--projection and discuss the stabilizing effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.

DOI
10.1137/s0097539796302452

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We propose a method for the stabilization of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM\cal SYMSYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting SYM\cal SYMSYM--projection and discuss the stabilizing effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.
We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM\cal SYMSYM, the symmetric subspace of the full state space of RRR redundant copies of the computer. We describe an efficient algorithm and quantum network effecting SYM\cal SYMSYM--projection and discuss the stabilising effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.
To implement quantum algorithms on a quantum computer, we must overcome the twin problems of fault-tolerance -- how can we realize a relatively noiseless computation by cleverly combining noisy components? -- and compilation -- how can we realize an arbitrary quantum algorithm given the basic operations available on the quantum device at hand? We show how treating the former problem via error-correcting codes enables greater flexibility in resolving the latter. Specifically, we explicitly leverage the fact that error-correcting codes introduce redundancy which renders physically distinct operators logically indistinguishable. In terms of computation, it suffices to implement any operator logically equivalent to some target, yet from a compilation perspective, certain choices may be preferable to others. Our novel contribution is making this intuition precise in the general setting of the special unitary group. In particular, we describe how to reduce the problem of making a compilation-ideal choice to a least squares problem and provide a closed form solution thereof. Using our framework, it is possible to circumvent inserting costly swaps to adhere to hardware connectivity; instead, we could realize the logical target through a distinct physical Hamiltonian that is natively accessible. We elucidate our approach using the [[4,2,2]][[4,2,2]][[4,2,2]] code. We discuss connections to compressed sensing that may pave the way to efficient compilation leveraging physical degrees of freedom.
We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an e#cient algorithm and quantum network e#ecting SYM--projection and discuss the stabilising e#ect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed. 1 Introduction Any realistic model of computation must conform to certain requirements imposed not by the mathematical properties of the model but by the laws of physics. Computations which require an exponentially increasing precision or exponential amount of time, space, energy or any other physical resource are normally regarded as unrealistic and intractable. Any actual computational process is subject to unavoidab...
Quantum state purification is the functionality that, given multiple copies of an unknown state, outputs a state with increased purity. This will be an essential building block for near- and middle-term quantum ecosystems before the availability of full fault tolerance, where one may want to suppress errors not only in expectation values but also in quantum states. We propose an effective state purification gadget with a moderate quantum overhead by projecting MMM noisy quantum inputs to their symmetric subspace defined by a set of projectors forming a symmetric subgroup with order MMM. Our method, applied in every short evolution over MMM redundant copies of noisy states, can suppress both coherent and stochastic errors by a factor of 1/M1/M1/M, respectively. This reduces the circuit implementation cost MMM times smaller than the state projection to the full symmetric subspace proposed by Barenco et al. more than two decades ago. We also show that our gadget purifies the depolarised inputs with probability ppp to asymptotically O(p2)O\left(p^{2}\right)O(p2) with an optimal choice of MMM when ppp is small. The sampling cost scales O(p−1)O\left(p^{-1}\right)O(p−1) for small ppp, which is also shown to be asymptotically optimal. Our method provides flexible choices of state purification depending on the hardware restrictions before fully fault-tolerant computation is available.
Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design.
All free-space quantum communications require the use of a quantum channel, which transports quantum bits in such a way that the quantum mechanical states of the qubits remain preserved from one end of the channel to the other one. In quantum computing the classical error coding methods could not be used, however we can construct a classical channel with zero redundancy error correction for any unitary channel. In our basically new quantum error correction approach, the classical states are coded into the eigenvectors and unitary transformations. In this paper, we show that with our new algorithm it’s possible to create redundancy-free quantum error correction. We also consider the redundancy-free implementation of a unitary error correcting operator. Our protocol achieves the redundancy-free quantum communication using local unitary operations and unitary matrices. These solutions could be useful for the free-space quantum communication.