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arXiv:2209.06673 · Physical Review A

Fault-tolerant preparation of quantum polar codes encoding one logical qubit

Ashutosh Goswami, Mehdi Mhalla, Valentin Savin

Published Oct 11, 2023• 6 citations
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Abstract

This paper explores an approach to fault-tolerant quantum computing (FTQC), relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane type, encoding one logical qubit, which we refer to as ${\mathcal{Q}}_{1}$ codes. First, we show that a subfamily of ${\mathcal{Q}}_{1}$ codes is equivalent to the well-known family of Shor codes. Moreover, we show that ${\mathcal{Q}}_{1}$ codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the fault-tolerant preparation of ${\mathcal{Q}}_{1}$ code states. We give a recursive procedure to prepare a ${\mathcal{Q}}_{1}$ code state, based on two-qubit Pauli measurements only. The procedure is not by itself fault-tolerant; however, the measurement operations therein provide redundant classical bits, which can be advantageously used for error detection. Fault tolerance is then achieved by combining the proposed recursive procedure with an error detection method. Finally, we consider the fault-tolerant error correction of ${\mathcal{Q}}_{1}$ codes. We use Steane error correction, which incorporates the proposed fault-tolerant code state preparation procedure. We provide numerical estimates of the logical error rates for ${\mathcal{Q}}_{1}$ and Shor codes of length 16 and 64 qubits, assuming a circuit-level depolarizing noise model. Remarkably, the ${\mathcal{Q}}_{1}$ code of length 64 qubits achieves a logical error rate very close to ${10}^{\ensuremath{-}6}$ for the physical error rate $p={10}^{\ensuremath{-}3}$, therefore demonstrating the potential of the proposed polar-codes-based approach to FTQC.

DOI
10.1103/physreva.108.042605

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This paper explores an approach to fault-tolerant quantum computing (FTQC), relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane type, encoding one logical qubit, which we refer to as Q1{\mathcal{Q}}_{1}Q1​ codes. First, we show that a subfamily of Q1{\mathcal{Q}}_{1}Q1​ codes is equivalent to the well-known family of Shor codes. Moreover, we show that Q1{\mathcal{Q}}_{1}Q1​ codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the fault-tolerant preparation of Q1{\mathcal{Q}}_{1}Q1​ code states. We give a recursive procedure to prepare a Q1{\mathcal{Q}}_{1}Q1​ code state, based on two-qubit Pauli measurements only. The procedure is not by itself fault-tolerant; however, the measurement operations therein provide redundant classical bits, which can be advantageously used for error detection. Fault tolerance is then achieved by combining the proposed recursive procedure with an error detection method. Finally, we consider the fault-tolerant error correction of Q1{\mathcal{Q}}_{1}Q1​ codes. We use Steane error correction, which incorporates the proposed fault-tolerant code state preparation procedure. We provide numerical estimates of the logical error rates for Q1{\mathcal{Q}}_{1}Q1​ and Shor codes of length 16 and 64 qubits, assuming a circuit-level depolarizing noise model. Remarkably, the Q1{\mathcal{Q}}_{1}Q1​ code of length 64 qubits achieves a logical error rate very close to 10\ensuremath−6{10}^{\ensuremath{-}6}10\ensuremath−6 for the physical error rate p=10\ensuremath−3p={10}^{\ensuremath{-}3}p=10\ensuremath−3, therefore demonstrating the potential of the proposed polar-codes-based approach to FTQC.
A fault-tolerant way to prepare logical code states of Q1{\mathcal{Q}}_{1}Q1​ codes, i.e., quantum polar codes encoding one qubit, was recently proposed. The fault tolerance therein is guaranteed by an error detection gadget, where, if an error is detected during the preparation, one entirely discards the preparation. Due to error detection, the preparation is probabilistic and its success rate, referred to as the preparation rate, decreases rapidly with the code length, preventing the preparation of code states of large code- lengths. In this paper, to improve the preparation rate, we consider a factory preparation of Q1{\mathcal{Q}}_{1}Q1​ code states, where one attempts to prepare several copies of Q1{\mathcal{Q}}_{1}Q1​ code states in parallel. Using an extra scheduling step, we can avoid discarding the preparation entirely every time an error is detected, hence, achieving an increased preparation rate in turn. We further provide a theoretical method to estimate preparation and logical error rates of Q1{\mathcal{Q}}_{1}Q1​ codes, prepared using factory preparation, which is shown to tightly fit the Monte Carlo simulation-based numerical results. Therefore, our theoretical method is useful for providing estimates for large code lengths, where Monte Carlo simulations are practically not feasible. Our numerical results, for a circuit-level depolarizing noise model, indicate that the preparation rate increases significantly, especially for large code-length NNN. For example, for N=256N=256N=256, it increases from 0.020.02%0.02 to 2727%27 for a practically interesting physical error rate $p={10}^{\ensuremath{-}3}$. Remarkably, a Q1{\mathcal{Q}}_{1}Q1​ code with N=256N=256N=256 achieves logical error rates around ${10}^{\ensuremath{-}11}$ and ${10}^{\ensuremath{-}15}$ for $p={10}^{\ensuremath{-}3}$ and $p=3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, respectively. This corresponds to an improvement of about three orders of magnitude compared to a surface code with similar code length and minimum distance, thus showing the promise of the proposed scheme for large-scale fault-tolerant quantum computing.
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.
A fault tolerant way to prepare logical code-states of Q1\mathcal{Q}_{1}Q1​ codes, i.e., quantum polar codes encoding one qubit, has been recently proposed. The fault tolerance is guaranteed with the help of an error detection gadget, where if an error is detected, one declares a preparation failure and discards entirely the preparation. This makes the preparation probabilistic, whose success rate is referred to as the preparation rate. In this work, we consider a factory preparation procedure, where one attempts to prepare several copies of Q1\mathcal{Q}_{1}Q1​ code-states in parallel. With the help of an extra scheduling step, we can avoid discarding the preparation entirely, every time an error is detected, hence, we may achieve an increased preparation rate. Our numerical simulations, for a circuit-level depolarizing model, suggest that the preparation rate increases significantly, especially for large code-lengths N. For example, for N=256, it increases from 2% to 26% for a practically interesting physical error rate of p=10−310^{-3}10−3. Remarkably, a Q1\mathcal{Q}_{1}Q1​ code of length N=256 achieves logical error rates around 10−1010^{-10}10−10 and 10−1510^{-15}10−15, for physical error rates of p=10−310^{-3}10−3 and p=3×10−410^{-4}10−4, respectively, thus showing that factory preparation may allow to access the error protection provided by the larger Q1\mathcal{Q}_{1}Q1​ code-states.
Fault-tolerant logical operations for qubits encoded by Calderbank-Shor-Steane codes are discussed, with emphasis on methods that apply to codes of high rate, encoding kkk qubits per block with k>1k>1k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into XXX and ZZZ parts. Optimized methods to move logical qubits around and to achieve controlled-NOT and Toffoli gates are discussed. It is found that the number of time steps required to complete a fault-tolerant quantum computation is the same when k>1k>1k>1 as when k=1k=1k=1.
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.
Modularization promises to break down the design and implementation complexity of large scale quantum processors into smaller manageable subtasks. In this approach, quantum channels, realized for instance through physical rerouting of qubits or quantum teleportation, connect multiple modules. Each of those modules hosts a subset of qubits, e.g. multiple logical qubits, and provides quantum operations on them. In this work, we implement for the first time all logical operations required for modular fault-tolerant universal quantum computing with a non-Calderbank-Shor-Steane (CSS) code, the perfect [[5,1,3]][[5, 1, 3]][[5,1,3]] code, on a trapped-ion quantum computer. This code is the smallest quantum error-correcting (QEC) code capable of correcting any single-qubit error, making it a compact alternative to larger CSS codes. We demonstrate logical state teleportation and a full suite of fault-tolerant operations required for universal logical control, including logical state preparation, QEC with real-time feedback, logical measurements, magic-state preparation, logical entangling operations, and magic-state injection. Moreover, we characterize the logical spectator error picked up by idling logical qubits during quantum operations on distinct qubit registers and demonstrate a logical Pauli quantum process tomography that minimizes required sampling resources for logical tomography.