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arXiv:quant-ph/0002039 · Physical Review A

Methodology for quantum logic gate construction

Xinlan Zhou, Debbie Leung, Isaac L. Chuang

Published Oct 18, 2000• 350 citations
arXivHTMLPDF

Abstract

We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation [Gottesman and Chuang, Nature (London) 402, 390 (1999)] and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the $\ensuremath{\pi}/8$ and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed.

DOI
10.1103/physreva.62.052316

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We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation [Gottesman and Chuang, Nature (London) 402, 390 (1999)] and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the \ensuremathπ/8\ensuremath{\pi}/8\ensuremathπ/8 and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed.
We present a general method to construct quantum logic gates using one-bit``teleportation'' as a basic primitive, extending previous results based ontraditional two-bit teleportation (D. Gottesman and I. L. Chuang, Nature {\\bf402}, 390, 1999). This allows the realization of a variety of useful quantumoperations that cannot be directly performed due to some physical constraints.In particular, this technique leads to straightforward and systematicconstruction of many fault-tolerant encoded operations, including the pi/8\\pi/8pi/8and Toffoli gates, and remote quantum operations.
We present a general method to construct a universal set of quantum gates using probabilistic teleportation as a basic primitive. The technique generalizes the teleportation method of gate construction to partially entangled quantum channels. Without recourse to local filtering or entanglement concentration, using local rotation and CNOT operations followed by measurements in the computational basis, one can construct many encoded quantum operations with unit fidelity but less than unit probability. The technique can also be applied to the construction of remote quantum gates that cannot be directly performed.
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.
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.
ABSTRACT The quantum gate teleportation mechanism allows for the fault‐tolerant implementation of “Clifford hierarchies” of gates assuming, among other things, a fault‐tolerant implementation of the Pauli gates. We discuss how this method can be extended to assume the fault‐tolerant implementation of any orthogonal unitary basis of operators, in such a way that each of these bases can be used to implement its own hierarchy of gates. Noticing that all of these hierarchies are defined through the same algebraic construction, we study this construction from a group‐theoretic perspective. This allows us to deduce general results concerning all the hierarchies. We also discuss alternative hierarchies, study their properties, and compute some of their first few levels.