Wigner Function Negativity and Contextuality in Quantum Computation on Rebits

Delfosse, Nicolas; Guérin, Philippe; Bian, Jacob; Raussendorf, Robert

doi:10.1103/physrevx.5.021003

Cited by 178 publications

(252 citation statements)

References 32 publications

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“…[24,25] for details. The restriction is that all Pauli operators to be measured are either X type or Z type, and the measurements cannot be adaptive.…”

Section: Discussion and Previous Workmentioning

confidence: 99%

“…To enable a comparison between Theorem 4 and the results of Ref. [24], one can employ a discrete Wigner function representation of stabilizer states and Clifford operations on qubits developed by Delfosse et al [25]. The latter is applicable only to states with real amplitudes and to Clifford operations that do not mix X-type and Z-type Pauli operators (CSS-preserving operations).…”

Section: Discussion and Previous Workmentioning

confidence: 99%

“…A preliminary analysis shows that combining the results of Refs. [24,25] yields a classical algorithm for simulating a restricted class of PBC on n qubits in time M 2n polyðnÞ≈2 0.543n polyðnÞ, where M ¼2…”

Section: Discussion and Previous Workmentioning

confidence: 99%

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Trading Classical and Quantum Computational Resources

2016

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We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First, we consider sparse quantum circuits such that each qubit participates in Oð1Þ two-qubit gates. It is shown that any sparse circuit on n þ k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time 2 OðkÞ polyðnÞ. Second, we study Pauli-based computation (PBC), where allowed operations are nondestructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n þ k qubits can be simulated by PBCs on n qubits and a classical processing that takes time 2OðkÞ polyðnÞ. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time 2 αn polyðnÞ, where α ≈ 0.94. This improves upon the brute-force simulation method, which takes time 2 n polyðnÞ. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.

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“…[24,25] for details. The restriction is that all Pauli operators to be measured are either X type or Z type, and the measurements cannot be adaptive.…”

Section: Discussion and Previous Workmentioning

confidence: 99%

Section: Discussion and Previous Workmentioning

confidence: 99%

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Trading Classical and Quantum Computational Resources

2016

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“…Quasi-probability distributions have been previously used to construct classical algorithms for simulation of quantum circuits [17,18]. Our work can be viewed as an application of these methods to the problem of simulating ideal quantum circuits by noisy ones.…”

Section: Examplesmentioning

confidence: 99%

Error Mitigation for Short-Depth Quantum Circuits

Temme¹,

Bravyi²,

Gambetta³

2017

Phys. Rev. Lett.

1,073

987

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Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and don't require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasi-probability distribution.From the time quantum computation generated wide spread interest, the strongest objection to its viability was the sensitivity to errors and noise. In an early paper, William Unruh [1] found that the coupling to the environment sets an ultimate time and size limit for any quantum computation. This initially curbed the hopes that the full advantage of quantum computing could be harnessed, since it set limits on the scalability of any algorithm. This problem was, at least in theory, remedied with the advent of quantum error correction [2][3][4]. It was proven that if both the decoherence and the imprecision of gates could be reduced below a finite threshold value, then quantum computation could be performed indefinitely [5,6]. Although it is the ultimate goal to reach this threshold in an experiment that is scalable to larger sizes, the overhead that is needed to implement a fully fault-tolerant gate set with current codes [7] seems prohibitively large [8,9]. In turn, it is expected that in the near term the progress in quantum experiments will lead to devices with dynamics, which are beyond what can be simulated with a conventional computer. This leads to the question: what computational tasks could be accomplished with only limited, or no error correction?The suggestions of near-term applications in such quantum devices mostly center around quantum simulations with short-depth circuit [10][11][12] and approximate optimization algorithms [13]. Furthermore, certain problems in material simulation may be tackled by hybrid quantum-classical algorithms [14]. In most such applications, the task can be abstracted to applying a short-depth quantum circuits to some simple initial state and then estimating the expectation value of some observable after the circuit has been applied. This estimation must be accurate enough to achieve a simulation precision comparable or exceeding that of classical algorithms. Yet, although the quantum system evolves coherently for the most part of the short-depth circuit, the effects of decoherence already become apparent...

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“…It means that any quantum state can be fully measured given sufficiently many copies. The rebit scheme [6], for example, does not satisfy this.One of our results is that for any number n of qubits there exists a QCSI scheme that satisfies both conditions (C1) and (C2). The reason why both conditions can simultaneously hold lies in a fundamental distinction between observables that can be measured directly in a given qubit QCSI scheme from those that can only be inferred by measurement of other observables.…”

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confidence: 99%

Contextuality as a Resource for Models of Quantum Computation with Qubits

et al. 2017

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A central question in quantum computation is to identify the resources that are responsible for quantum speed-up. Quantum contextuality has been recently shown to be a resource for quantum computation with magic states for odd-prime dimensional qudits and two-dimensional systems with real wave functions. The phenomenon of state-independent contextuality poses a priori an obstruction to characterizing the case of regular qubits, the fundamental building block of quantum computation. Here, we establish contextuality of magic states as a necessary resource for a large class of quantum computation schemes on qubits. We illustrate our result with a concrete scheme related to measurement-based quantum computation. DOI: 10.1103/PhysRevLett.119.120505 The model of quantum computation by state injection (QCSI) [1] is a leading paradigm of fault-tolerance quantum computation. Therein, quantum gates are restricted to belong to a small set of classically simulable gates, called Clifford gates [2], that admit simple fault-tolerant implementations [3]. Universal quantum computation is achieved via injection of magic states [1], which are the source of quantum computational power of the model.A central question in QCSI is to characterize the physical properties that magic states need to exhibit in order to serve as universal resources. In this regard, quantum contextuality has recently been established as a necessary resource for QCSI. This was first achieved for quopit systems [4,5], where the local Hilbert space dimension is an odd prime power, and subsequently for local dimension two with the case of rebits [6]. In the latter, the density matrix is constrained to be real at all times.In this Letter we ask "Can contextuality be established as a computational resource for QCSI on qubits?" This is not a straightforward extension of the quopit case because the multiqubit setting is complicated by the presence of stateindependent contextuality among Pauli observables [7,8]. Consequently, every quantum state of n ≥ 2 qubits is contextual with respect to Pauli measurements, including the completely mixed one [5]. It is thus clear that contextuality of magic states alone cannot be a computational resource for every QCSI scheme on qubits.Yet, there exist qubit QCSI schemes for which contextuality of magic states is a resource, and we identify them in this Letter. Specifically, we consider qubit QCSI schemes M O that satisfy the following two constraints: (C1) Resource character. There exists a quantum state that does not exhibit contextuality with respect to measurements available in M O . (C2) Tomographic completeness. For any state ρ, the expectation value of any Pauli observable can be inferred via the allowed operations of the scheme.The motivation for these constraints is the following. Condition (C1) constitutes a minimal principle that unifies, simplifies and extends the quopit [5] and rebit [6] settings. While seemingly a weak constraint, it excludes the possibility of Mermin-type state-independent contextuality [7,8] amo...

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Wigner Function Negativity and Contextuality in Quantum Computation on Rebits

Cited by 178 publications

References 32 publications

Trading Classical and Quantum Computational Resources

Trading Classical and Quantum Computational Resources

Error Mitigation for Short-Depth Quantum Circuits

Contextuality as a Resource for Models of Quantum Computation with Qubits

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