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arXiv:1303.6971 · Physical Review A

Composite Toffoli gate with two-round error detection

Cody Jones

Published May 31, 2013• 24 citations
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Abstract

We introduce a fault-tolerant construction to implement a composite quantum operation of four overlapping Toffoli gates. The same construction can produce two independent Toffoli gates. This result lowers resource overheads in designs for quantum computers by more than an order of magnitude. The procedure uses Clifford operations and 64 copies of the non-Clifford gate $T=\mathrm{exp}[i\ensuremath{\pi}(I\ensuremath{-}{\ensuremath{\sigma}}^{z})/8]$. Quantum codes detect errors in the circuit. When the dominant source of error is $T$-gate failure with probability $p$, then the composite Toffoli circuit has a postselected failure rate of $3072{p}^{4}$ to lowest order.

DOI
10.1103/physreva.87.052334

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IEEE International Symposium on Multiple-Valued Logic (ISMVL)•May 1, 2016•18 citations
We introduce a fault-tolerant construction to implement a composite quantum operation of four overlapping Toffoli gates. The same construction can produce two independent Toffoli gates. This result lowers resource overheads in designs for quantum computers by more than an order of magnitude. The procedure uses Clifford operations and 64 copies of the non-Clifford gate T=exp[i\ensuremathπ(I\ensuremath−\ensuremathσz)/8]T=\mathrm{exp}[i\ensuremath{\pi}(I\ensuremath{-}{\ensuremath{\sigma}}^{z})/8]T=exp[i\ensuremathπ(I\ensuremath−\ensuremathσz)/8]. Quantum codes detect errors in the circuit. When the dominant source of error is TTT-gate failure with probability ppp, then the composite Toffoli circuit has a postselected failure rate of 3072p43072{p}^{4}3072p4 to lowest order.
We present two constructions for the Toffoli gate which substantially reduce resource costs in fault-tolerant quantum computing. The first contribution is a Toffoli gate requiring Clifford operations plus only four $T=\mathrm{exp}(i\ensuremath{\pi}{\ensuremath{\sigma}}^{z}/8)$ gates, whereas conventional circuits require seven TTT gates. An extension of this result is that adding nnn control inputs to a controlled gate requires 4n4n4n TTT gates, whereas the best prior result was 8n8n8n. The second contribution is a quantum circuit for the Toffoli gate which can detect a single ${\ensuremath{\sigma}}^{z}$ error occurring with probability ppp in any one of eight TTT gates required to produce the Toffoli gate. By postselecting circuits that did not detect an error, the posterior error probability is suppressed to lowest order from 4p4p4p (or 7p7p7p, without the first contribution) to 28p228{p}^{2}28p2 for this enhanced construction. In fault-tolerant quantum computing, this construction can reduce the overhead for producing logical Toffoli gates by an order of magnitude.
A non-Clifford gate is required for universal quantum computation, and, typically, this is the most error-prone and resource-intensive logical operation on an error-correcting code. Small, single-qubit rotations are popular choices for this non-Clifford gate, but certain three-qubit gates, such as Toffoli or controlled-controlled-ZZZ (ccz), are equivalent options that are also more suited for implementing some quantum algorithms, for instance, those with coherent classical subroutines. Here, we calculate error rates and resource overheads for implementing logical ccz with pieceable fault tolerance, a nontransversal method for implementing logical gates. We provide a comparison with a nonlocal magic-state scheme on a concatenated code and a local magic-state scheme on the surface code. We find the pieceable fault-tolerance scheme particularly advantaged over magic states on concatenated codes and in certain regimes over magic states on the surface code. Our results suggest that pieceable fault tolerance is a promising candidate for fault tolerance in a near-future quantum computer.
Efficient constructions for quantum logic are essential since quantum computation is experimentally challenging. This thesis develops quantum logic synthesis as a paradigm for reducing the resource overhead in fault-tolerant quantum computing. The model for error correction considered here is the surface code. After developing the theory behind general logic synthesis, the resource costs of magic-state distillation for the T=exp⁡(iπ(I−Z)/8)T = \exp(i π(I-Z)/8)T=exp(iπ(I−Z)/8) gate are quantitatively analyzed. The resource costs for a relatively new protocol distilling multi-qubit Fourier states are calculated for the first time. Four different constructions of the fault-tolerant Toffoli gate, including two which incorporate error detection, are analyzed and compared. The techniques of logic synthesis reduce the cost of fault-tolerant quantum computation by one to two orders of magnitude, depending on which benchmark is used. Using resource analysis for TTT gates and Toffoli gates, several proposals for constructing arbitrary quantum gates are compared, including "Clifford+TTT" sequences, VVV-basis sequences, phase kickback, and programmable ancilla rotations. The application of arbitrary gates to quantum algorithms for simulating chemistry is discussed as well. Finally, the thesis examines the techniques which lead to efficient constructions of quantum logic, and these observations point to even broader applications of logic synthesis.
A fully fault-tolerant implementation of the quantum error-detecting Iceberg [[2m,2m−2,2]][[2m, 2m-2, 2]][[2m,2m−2,2]] code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.
Building generic gates from a restricted gate set is a difficult but important problem, especially in the noisy regime where only a limited set of noise-resistant gates are available, e.g., fault-tolerant Clifford gates (generated by Hadamard, phase, and gates) and fault-tolerant Toffoli gates. The Toffoli is also often used as a building block of many algorithms and will need to be constructed if not directly available. This makes Clifford + Toffoli an attractive gate set for building generic gates. In this Letter we give a simple and efficient algorithm for building an approximate single-qubit rotation using only Clifford + Toffoli , which in turn enables any generic single-qubit unitary. An important difference compared to earlier attempts is that the use of the Toffoli allows us to use simple rounding as opposed to a complicated approximation algorithm. The resulting gate array does not rely on repeatedly applying a fixed rotation, but immediately applies a rotation R θ * that is ε -close to the desired rotation R θ , with a success probability strictly greater than 1 / 2 . It can be rerun upon failure, giving an expected number of repetitions strictly less than 2, an expected Toffoli count strictly less than 4 ⌈ log 1 ε ⌉ , an expected gate depth strictly less than 4 ⌈ log 1 ε ⌉ + 6 , and uses 2 ⌈ log 1 ε ⌉ ancillas. The small circuit depth of our construction enables low-noise gates on existing quantum computational devices, and allows for arbitrary precision using a very modest number of ancillas. Published by the American Physical Society 2024
The quest of achieving higher computing performance is driving the research on quantum computing, which is reporting new milestones almost on a daily basis. For practical quantum circuit design, fault tolerance is an essential condition. This is achieved by mapping the target functions into the Clifford+T group of elementary quantum gates. Furthermore, the application of error-correcting codes in quantum circuits requires the quantum gates to be formed between adjacent Qubits. In this work, we improve the state-of-the-art quantum circuit design by addressing both of the above challenges. First, we propose a novel mapping of Multiple-Control Toffoli (MCT) gates to Clifford+T group gates, which achieves lower gate count compared to earlier work. Secondly, we show a generic way to convert any Clifford+T circuit into a nearest neighbor one. We validate the efficacy of our approach with detailed experimental studies.