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arXiv:1509.05032 · Physical Review A

Reducing the overhead for quantum computation when noise is biased

Paul Webster, Stephen D. Bartlett, David Poulin

Published Dec 3, 2015• 44 citations
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Abstract

We analyze a model for fault-tolerant quantum computation with low overhead suitable for situations where the noise is biased. The basis for this scheme is a gadget for the fault-tolerant preparation of magic states that enable universal fault-tolerant quantum computation using only Clifford gates that preserve the noise bias. We analyze the distillation of $|T\ensuremath{\rangle}$-type magic states using this gadget at the physical level, followed by concatenation with the 15-qubit quantum Reed-Muller code, and comparing our results with standard constructions. In the regime where the noise bias (rate of Pauli $Z$ errors relative to other single-qubit errors) is greater than a factor of 10, our scheme has lower overhead across a broad range of relevant noise rates.

DOI
10.1103/physreva.92.062309

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We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.

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We analyze a model for fault-tolerant quantum computation with low overhead suitable for situations where the noise is biased. The basis for this scheme is a gadget for the fault-tolerant preparation of magic states that enable universal fault-tolerant quantum computation using only Clifford gates that preserve the noise bias. We analyze the distillation of ∣T\ensuremath⟩|T\ensuremath{\rangle}∣T\ensuremath⟩-type magic states using this gadget at the physical level, followed by concatenation with the 15-qubit quantum Reed-Muller code, and comparing our results with standard constructions. In the regime where the noise bias (rate of Pauli ZZZ errors relative to other single-qubit errors) is greater than a factor of 10, our scheme has lower overhead across a broad range of relevant noise rates.
Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of 3 × 10 −7 , using only 53 qubits and 5.5 error correction rounds, under a noise bias of η ≳ 5 × 10 6 and a phase-flip noise rate of 0.1%. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of η ≳ 80, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the X stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as unfolded distillation.
We analyze the resource overhead of recently proposed methods for universal fault-tolerant quantum computation using concatenated codes. Namely, we examine the concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, which allows for the construction of the 49- and 105-qubit codes that do not require the need for magic state distillation for universality. We compute a lower bound for the adversarial noise threshold of the 105-qubit code and find it to be $8.33\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$ We obtain a depolarizing noise threshold for the 49-qubit code of $9.69\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ which is competitive with the 105-qubit threshold result of $1.28\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. We then provide lower bounds on the resource requirements of the 49- and 105-qubit codes and compare them with the surface code implementation of a logical TTT gate using magic state distillation. For the sampled input error rates and noise model, we find that the surface code achieves a smaller overhead compared to our concatenated schemes.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state~\cite{Bravyi}. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4)210.2p^3+O(p^4)210.2p3+O(p4). At p=10−3p=10^{-3}p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−72.2\times10^{-7}2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.
We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.