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arXiv:quant-ph/0411206

Constructing arbitrary single-qubit fault-tolerant gates

Austin G. Fowler

Published Nov 30, 2004• 6 citations
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Abstract

We present a method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbtrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation, however, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the 7-qubit Steane code, it is shown that the longest practical gates sequences can be found. We also analyse the practicality of the fault-tolerant approximations of the phase rotation gates used in Shor’s algorithm and find that simple non-fault-tolerant phase rotations are more robust for realistic error rates. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined. In large-scale quantum computation, every qubit of data is encoded across multiple physical qubits to form a logical qubit permitting quantum error correction

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Fault-tolerant logical gate networks for Calderbank-Shor-Steane codes

Andrew M. Steane, Ben Ibinson

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 $k$ qubits per block with $k>1$. 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 $X$ and $Z$ 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 $k>1$ as when $k=1$.

arXiv:quant-ph/0311014•Physical Review A•Nov 30, 2005•44 citations
We present a method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbtrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation, however, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the 7-qubit Steane code, it is shown that the longest practical gates sequences can be found. We also analyse the practicality of the fault-tolerant approximations of the phase rotation gates used in Shor’s algorithm and find that simple non-fault-tolerant phase rotations are more robust for realistic error rates. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined. In large-scale quantum computation, every qubit of data is encoded across multiple physical qubits to form a logical qubit permitting quantum error correction
We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.
We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.
For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+ T set of base gates and compared them to when additionally including Z -rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced to T -counts by recursively applying a Z -rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to 54±3% when using the Z -rotation catalyst circuit approach and by up to 33±2% when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets of Z -rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
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.