Surface code implementation of block code state distillation

Fowler, Austin G.; Devitt, Simon J.; Jones, Cody

doi:10.1038/srep01939

Cited by 86 publications

(122 citation statements)

References 45 publications

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“…Furthermore, topological quantum codes, such as the surface code, enable a direct measurement of certain logical Pauli operators by measuring a properly chosen subset of physical qubits [12]. Several fault-tolerant protocols for preparing encoded magic states such as jHi have been developed [13][14][15][16][17]. PBCs implicitly appeared in the previous work on quantum fault tolerance.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Trading Classical and Quantum Computational Resources

2016

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“…The magic states must themselves be prepared using a fault tolerant protocol for "magic state distillation" [16], which is relatively resource intensive. For example, in the case of the surface code, the overhead associated with logical T gates exceeds that of any logical Clifford gate by orders of magnitude [17,18]. Thus it is likely that the first logical circuits demonstrated in the lab will be Clifford þ T circuits dominated by Clifford gates.…”

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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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“…The most common approach is that of magic states [5,6], encoded ancilla qubits that, combined with available transversal circuits (usually implementing Clifford operations), serve to complete a universal set of logical circuits (usually by implementing a T or Toffoli gate). Ideally, these magic states would be efficiently constructible themselves, but current so-called distillation procedures actually incur large overheads in terms of time and qubits [7,8].…”

Section: Introductionmentioning

confidence: 99%

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

2016

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It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.

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Surface code implementation of block code state distillation

Cited by 86 publications

References 45 publications

Trading Classical and Quantum Computational Resources

Trading Classical and Quantum Computational Resources

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

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