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arXiv:2307.15226 · Physical Review A

Factory-based fault-tolerant preparation of quantum polar codes encoding one logical qubit

Ashutosh Goswami, Mehdi Mhalla, Valentin Savin

Published Jul 12, 2024• 3 citations
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Abstract

A fault-tolerant way to prepare logical code states of ${\mathcal{Q}}_{1}$ 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 ${\mathcal{Q}}_{1}$ code states, where one attempts to prepare several copies of ${\mathcal{Q}}_{1}$ 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 ${\mathcal{Q}}_{1}$ 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 $N$. For example, for $N=256$, it increases from $0.02%$ to $27%$ for a practically interesting physical error rate $p={10}^{\ensuremath{-}3}$. Remarkably, a ${\mathcal{Q}}_{1}$ code with $N=256$ 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.

DOI
10.1103/physreva.110.012438

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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−3p={10}^{\ensuremath{-}3}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{10}^{\ensuremath{-}11}10\ensuremath−11 and 10\ensuremath−15{10}^{\ensuremath{-}15}10\ensuremath−15 for p=10\ensuremath−3p={10}^{\ensuremath{-}3}p=10\ensuremath−3 and p=3\ifmmode×\else\texttimes\fi10\ensuremath−4p=3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}p=3\ifmmode×\else\texttimes\fi10\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.
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.
One of the major challenges in realizing fault-tolerant quantum computers (FTQCs) is the requirement for a large number of physical qubits. To address this issue, high-rate quantum error correcting codes, which efficiently embed logical qubits into physical qubits, have recently attracted considerable attention. Among such codes, quantum BCH codes, which offer both high rates and large code distances, are promising yet underexplored candidates. However, no fault-tolerant ancilla preparation method specialized for this class had been established. We employ a two-stage approach (non-fault-tolerant preparation + entanglement distillation) for ancilla preparation. We then propose a framework for designing low-overhead distillation method that strategically leverages the cyclic symmetry of quantum BCH codes to determine which non-fault-tolerant circuits can successfully produce a fault-tolerant state. Numerical simulations on several high-performance quantum BCH codes up to 127 qubits demonstrate that our method achieves lower spatial overhead and logical error rates than conventional distillation circuits. Furthermore, we evaluated the logical error rates under a circuit-level noise model, and obtained performance benchmarks in realistic settings. This efficient state preparation technique is expected to contribute to the early realization of practical FTQCs, particularly on highly connected quantum platforms such as neutral atom systems.
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 ${10}^{\ensuremath{-}6}$ for the physical error rate $p={10}^{\ensuremath{-}3}$, therefore demonstrating the potential of the proposed polar-codes-based approach to FTQC.
In fault-tolerant quantum computation, the preparation of logical states is a ubiquitous subroutine, yet significant challenges persist even for the simplest states required. In the present work, we present a unitary, scalable, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes. Unlike previous unitary approaches whose fault-distance remains constant with increasing code distance, our scheme ensures that the protection offered by the code is preserved during state preparation. Building on strategies discovered by reinforcement learning for the surface-17 code, we generalize the construction to arbitrary code distances and both rotated and unrotated surface codes. The proposed encoding relies only on geometrically local gates, and is therefore fully compatible with planar 2D qubit connectivity, and it achieves circuit depth scaling as O(d)\mathcal{O}(d)O(d), consistent with fundamental entanglement-generation bounds. We design explicit stabilizer-expanding circuits with and without ancilla-mediated connectivity and analyze their error-propagation behavior. Numerical simulations under depolarizing noise show that our unitary encoding without ancillas outperforms standard stabilizer-measurement-based schemes, reducing logical error rates by up to an order of magnitude. These results make the scheme particularly relevant for platforms such as trapped ions and neutral atoms, where measurements are costly relative to gates and idling noise is considerably weaker than gate noise. Our work bridges the gap between measurement-based and unitary encodings of surface-code states and opens new directions for distance-preserving state preparation in fault-tolerant quantum computation.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.
Quantum computing has enormous potential, but this can only be realised if quantum errors can be controlled sufficiently to allow quantum algorithms to be completed reliably. However, quantum-error-corrected logical quantum bits (qubits) which can be said to have achieved meaningful error suppression have not yet been demonstrated. This thesis reports research on several topics related to the challenge of designing fault-tolerant quantum computers. The first topic is a proposal for achieving large-scale error correction with the surface code in a silicon donor based quantum computing architecture. This proposal relaxes some of the stringent requirements in donor placement precision set by previous ideas from the single atom level to the order of 10 nm in some regimes. This is shown by means of numerical simulation of the surface code threshold. The second topic then follows, it is the development of a method for benchmarking and assessing the performance of small error correcting codes in few-qubit systems, introducing a metric called 'integrity' - closely linked to the trace distance -- and a proposal for experiments to demonstrate various stepping stones on the way to 'strictly superior' quantum error correction. Most quantum error correcting codes, including the surface code, do not allow for fault-tolerant universal computation without the addition of extra gadgets. One method of achieving universality is through a process of distilling and then consuming high quality 'magic states'. This process adds additional overhead to quantum computation over and above that incurred by the use of the base level quantum error correction. The latter parts of this thesis report an investigation into how many physical qubits are needed in a `magic state factory' within a surface code quantum computer and introduce a number of techniques to reduce the overhead of leading magic state techniques. It is found that universal quantum computing is achievable with ∼ 16 million qubits if error rates across a device are kept below 10-4. In addition, the thesis introduces improved methods of achieving magic state distillation for unconventional magic states that allow for logical small angle rotations, and show that this can be more efficient than synthesising these operations from the gates provided by traditional magic states.