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arXiv:1805.12082

Depth reduction for quantum Clifford circuits through Pauli measurements

Yi-Cong Zheng, Ching–Yi Lai, Todd A. Brun, L. C. Kwek

Published May 30, 2018• 2 citations
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Abstract

Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|Φ^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, $O(1)$ CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by $O(1)$ steps of Steane syndrome extraction if clean CSS stabilizer states are available.

DOI
10.48550/arxiv.1805.12082

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Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an nnn-qubit Clifford circuit UUU can be implemented by preparing an ancillary stabilizer state (I⊗U)∣Φ+⟩⊗n(I\otimes U)|Φ^+\rangle^{\otimes n}(I⊗U)∣Φ+⟩⊗n for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing O(1)O(1)O(1) ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, O(1)O(1)O(1) CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by O(1)O(1)O(1) steps of Steane syndrome extraction if clean CSS stabilizer states are available.
Abstract Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode k > 1 qubits in n physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long in situ computation time. In this paper, we focus on Calderbank–Shor–Steane [[ n , k , d ]] (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitrary logical Clifford circuit can be implemented fault-tolerantly in O (1) steps in situ via either Knill or Steane syndrome measurement circuit, with the qualified ancilla states efficiently prepared. Particularly, for those codes satisfying k / n ∼ Θ(1), the resource scaling for Clifford circuits implementation on the logical level can be the same as on the physical level up to a constant, which is independent of code distance d . With a suitable pipeline to produce ancilla states, our scheme requires only a modest resource cost in physical qubits, physical gates, and computation time for very large scale FTQC.
Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to date. In this work, we prove some critical properties of Calderbank-Shor-Steane (CSS) codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates.
Stabilizer states are extensively studied in quantum information theory for their structures based on the Pauli group. Calderbank-Shor-Steane (CSS) stabilizer states are of particular importance in their application to fault-tolerant quantum computation (FTQC). However, how to fault-tolerantly prepare arbitrary CSS stabilizer states for general CSS stabilizer codes is still unknown, and their preparation can be highly costly in computational resources. In this paper, we show how to prepare a large class of CSS stabilizer states useful for FTQC. We propose distillation protocols using syndrome encoding by classical codes or quantum CSS codes. Along the same lines, we show that classical coding techniques can reduce the ancilla consumption in Steane syndrome extraction by using additional transversal controlled-not gates and classical computing power. In the scenario of a fixed ancilla consumption rate, we can increase the frequency of quantum error correction and effectively lower the error rate.
Simulation of stabilizer circuits is a well-studied problem in quantum information processing, with a number of highly optimized algorithms available. Yet, we argue that further improvements can arise from the theoretical structure of stabilizer operations themselves. We focus on the subclass of stabilizer circuits composed of Calderbank-Shor-Steane (CSS)-preserving stabilizer operations, which naturally appear in fault-tolerant computations over CSS stabilizer codes. Using elementary circuit transformation techniques, we show that such circuits can be exactly rewritten as classical probabilistic circuits that reproduce measurement statistics. This rewriting introduces no computational overhead, in contrast to the general case of stabilizer circuits. To clarify the origin of this simplification, we introduce the standard quadratic form representation of general stabilizer operations (Clifford channels). It provides an efficient way to describe compositions of stabilizer operations and thus to simulate stabilizer circuits. CSS-preserving operations correspond to purely linear forms, which under a Walsh-Hadamard-Fourier transform yield a noncontextual hidden variable model, providing an alternative proof of the introduced rewriting. Finally, we develop a theory of reference frames for multiqubit systems, where frames are encoded by quadratic forms. This allows us to express stabilizer operations as probabilistic maps for proper reference frames. Non-CSS-preserving stabilizer circuits require dynamical modifications of reference frames, embodying a contextuality resource that leads to the computational overhead. This framework provides a new perspective on simulating stabilizer and near-stabilizer circuits within dynamically evolving quasiprobability models.
In this letter, we introduce a method to synthesize an nnn-qubit Clifford unitary CCC from the stabilizer tableau of its inverse C†C†C†, using ancilla qubits and measurements. The procedure uses ancillary ∣+⟩|+\rangle∣+⟩ states, controlled-Paulis, XXX-basis measurements and single-qubit Pauli corrections on the data qubits (based on the measurement results). This introduces a new normal form for Clifford synthesis, with the number of two-qubit gates required exactly equal to the weight of the stabilizer tableau, and a depth linear in nnn.
Given that quantum error correction processes are unreliable, an efficient error syndrome extraction circuit should use fewer ancillary qubits, quantum gates, and measurements, while maintaining low circuit depth, to minimizing the circuit area, roughly defined as the product of circuit depth and the number of physical qubits. We propose to design parallel flagged syndrome extraction with shared flag qubits for quantum stabilizer codes. Versatile parallelization techniques are employed to minimize the required circuit area, thereby improving the error threshold and overall performance. Specifically, measurement outcomes across multiple rounds of syndrome extraction are integrated into a lookup table decoder, enabling parallelization of multiple stabilizer measurements with shared flag qubits. Additionally, we introduce an adaptive technique to reduce the overhead from excessive syndrome extraction. We present flag-sharing and fully parallel schemes for the[ ⁣[17,1,5] ⁣][\![17,1,5]\!][[17,1,5]],[ ⁣[19,1,5] ⁣][\![19,1,5]\!][[19,1,5]]Calderbank-Shor-Steane (CSS) codes and the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]non-CSS code, where the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]implementation achieves the minimum known circuit area. Numerical simulations have demonstrated improved pseudo-thresholds for these codes by up to an order of magnitude compared to previous schemes in the literature.