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arXiv:2003.12328 · Quantum Science and Technology

Constant depth fault-tolerant Clifford circuits for multi-qubit large block codes

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

Published Jul 6, 2020• 11 citations
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Abstract

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.

DOI
10.1088/2058-9565/aba34d

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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.
Fault-tolerant quantum computation (FTQC) schemes that use multiqubit large block codes can potentially reduce the resource overhead to a great extent. A major obstacle is the requirement for a large number of clean ancilla states of different types without correlated errors inside each block. These ancilla states are usually logical stabilizer states of the data-code blocks, which are generally difficult to prepare if the code size is large. Previously, we have proposed an ancilla distillation protocol for Calderbank-Shor-Steane (CSS) codes by classical error-correcting codes. It was assumed that the quantum gates in the distillation circuit were perfect; however, in reality, noisy quantum gates may introduce correlated errors that are not treatable by the protocol. In this paper, we show that additional postselection by another classical error-detecting code can be applied to remove almost all correlated errors. Consequently, the revised protocol is fully fault tolerant and capable of preparing a large set of stabilizer states sufficient for FTQC using large block codes. At the same time, the yield rate can be boosted from $O({t}^{\ensuremath{-}2})$ to O(1)O(1)O(1) in practice for an [[n,k,d=2t+1]][[n,k,d=2t+1]][[n,k,d=2t+1]] CSS code. Ancilla preparation for the [[23,1,7]] quantum Golay code is numerically studied in detail through Monte Carlo simulation. The results support the validity of the protocol when the gate failure rate is reasonably low. To the best of our knowledge, this approach is one of the first attempts to prepare general large block stabilizer states free of correlated errors for FTQC in a fault-tolerant and efficient manner.
Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.
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.
Flagging the Clifford hierarchy:~Fault-tolerant logical π2l\fracπ{2^l}2lπ​ rotations via measuring circuit gauge operators of non-Cliffords
We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant RZ‾(π2l)R_{\overline{Z}}(\fracπ{2^l})RZ​(2lπ​) gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with O(l)O(l)O(l) gates and ancillae which implement fault-tolerant logical RZ(π2l)R_{Z}(\fracπ{2^l})RZ​(2lπ​) or RZZ(π2l)R_{ZZ}(\fracπ{2^l})RZZ​(2lπ​) gates on any [[k+2,k,2]][[k + 2, k, 2]][[k+2,k,2]] iceberg code and fault-tolerant circuits of size O(l)O(l)O(l) for preparing ∣π2l⟩|\fracπ{2^l}\rangle∣2lπ​⟩ resource states in the [[7,1,3]][[7,1,3]][[7,1,3]] code, which can be used to perform fault-tolerant RZ‾(π2l)R_{\overline{Z}}(\fracπ{2^l})RZ​(2lπ​) rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when lll is small relative to the precision required. We show how the circuits above can be generalized to π(x0.x1x2…xl)=∑jlπxj2jπ( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}π(x0​.x1​x2​…xl​)=∑jl​π2jxj​​ rotations with identical overheads in lll, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to 333 in the Steane code and how to increase the fault-distance of the π2\fracπ{2}2π​ iceberg circuit to 444 through concatenation in two-level iceberg codes. This yields a targeted logical RZ‾(π2)R_{\overline{Z}}(\fracπ{2})RZ​(2π​) gate with fault distance 444 on any row of logical qubits in an [[(k2+2)(k1+2),k1k2,4]][[(k_2+2)(k_1+2), k_1k_2, 4]][[(k2​+2)(k1​+2),k1​k2​,4]] code.
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.
Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes requires only three ancilla qubits, even with noisy Clifford gates. We compare the physical qubit and gate cost of our scheme to the magic state distillation protocol of Meier, Eastin, and Knill (MEK), which is efficient and uses a small stabilizer circuit. For low enough noise rates, we show that in some regimes the overhead can be improved by several orders of magnitude compared to the MEK scheme which uses Clifford operations encoded in the codes considered in this work.