Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities

Pashayan, Hakop; Wallman, Joel J.; Bartlett, Stephen D.

doi:10.1103/physrevlett.115.070501

Cited by 186 publications

(272 citation statements)

References 18 publications

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“…As was pointed out by Pashayan, Wallman, and Bartlett [24], such methods can be combined with Monte Carlo sampling techniques to enable classical simulation of general quantum circuits with the running time scaling exponentially with the quantity related to the negativity of the Wigner function. To enable a comparison between Theorem 4 and the results of Ref.…”

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%

“…[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%

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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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Section: Discussion and Previous Workmentioning

confidence: 99%

Section: Discussion and Previous Workmentioning

confidence: 99%

Section: Discussion and Previous Workmentioning

confidence: 99%

“…[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%

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Trading Classical and Quantum Computational Resources

2016

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“…Using distributed computation it is possible to simulate 40 qubit circuits [2]. For certain restricted classes of quantum circuits it is possible to do much better [3][4][5][6][7]. Most significantly, the GottesmanKnill theorem allows efficient classical simulation of quantum circuits composed of gates in the so-called Clifford group [3].…”

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confidence: 99%

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

Bravyi¹,

2016

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We present a new algorithm for classical simulation of quantum circuits over the Clifford þ T gate set. The runtime of the algorithm is polynomial in the number of qubits and the number of Clifford gates in the circuit but exponential in the number of T gates. The exponential scaling is sufficiently mild that the algorithm can be used in practice to simulate medium-sized quantum circuits dominated by Clifford gates. The first demonstrations of fault-tolerant quantum circuits based on 2D topological codes are likely to be dominated by Clifford gates due to a high implementation cost associated with logical T gates. Thus our algorithm may serve as a verification tool for near-term quantum computers which cannot in practice be simulated by other means. To demonstrate the power of the new method, we performed a classical simulation of a hidden shift quantum algorithm with 40 qubits, a few hundred Clifford gates, and nearly 50 T gates. DOI: 10.1103/PhysRevLett.116.250501 The path towards building a large-scale quantum computer will inevitably require verification and validation of small quantum devices. One way to check that such a device is working properly is to simulate it on a classical computer. This becomes impractical at some point because the cost of classical simulation typically grows exponentially with the size of a quantum system. With this fundamental limitation in mind it is natural to ask how well we can do in practice.Simulation methods which store a complete description of an n-qubit quantum state as a complex vector of size 2 n are limited to a small number of qubits n ≈ 30 − 40. For example, a state-of-the art implementation has been used to simulate Shor's factoring algorithm with 31 qubits and roughly half a million gates [1]. Using distributed computation it is possible to simulate 40 qubit circuits [2]. For certain restricted classes of quantum circuits it is possible to do much better [3][4][5][6][7]. Most significantly, the GottesmanKnill theorem allows efficient classical simulation of quantum circuits composed of gates in the so-called Clifford group [3]. In practice this allows one to simulate such circuits with thousands of qubits [1,4]. It also means that a quantum computer will need to use gates outside of the Clifford group in order to achieve useful speedups over classical computation. The full power of quantum computation can be recovered by adding a single non-Clifford gate to the Clifford group. A simple choice is the single-qubit T ¼ j0ih0j þ e iπ=4 j1ih1j gate. The Clifford þ T gate set obtained in this way is a natural instruction set for smallscale fault-tolerant quantum computers based on the surface code [8,9], and has been at the center of a recent renaissance in classical techniques for compiling quantum circuits [10][11][12].When it comes to realizing a logical (encoded) circuit, non-Clifford gates pose a serious challenge for any faulttolerant scheme based on 2D stabilizer codes [8,13] due to the lack of topological protection [14,15]. Such nonClifford gates can be imp...

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