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arXiv:2008.02631 · Physical Review A

Spin-orbit implementation of the Solovay-Kitaev decomposition of single-qubit channels

M. H. M. Passos, A. de Oliveira, M. C. de Oliveira, Antonio Zelaquett Khoury, J. A. O. Huguenin

Published Dec 3, 2020• 6 citations
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Abstract

The Solovay-Kitaev theorem allows us to approximate any single-qubit gate to arbitrary accuracy with a finite sequence of fundamental operations from a universal set of gates. Inspired by this decomposition, we present a quantum channel simulator capable of implementing any completely positive trace-preserving map. Our realization consists of one ancillary qubit, encoded in the transverse mode of a laser beam (orbital degree of freedom), one qubit system, encoded in its polarization (spin), one spin-orbit CNOT gate, and four single-qubit operations performed with prisms and polarization components. Our results describe the implementation of arbitrary single-qubit channels on the photon polarization using the transverse mode as the ancillary qubit.

DOI
10.1103/physreva.102.062601

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The Solovay-Kitaev theorem allows us to approximate any single-qubit gate to arbitrary accuracy with a finite sequence of fundamental operations from a universal set of gates. Inspired by this decomposition, we present a quantum channel simulator capable of implementing any completely positive trace-preserving map. Our realization consists of one ancillary qubit, encoded in the transverse mode of a laser beam (orbital degree of freedom), one qubit system, encoded in its polarization (spin), one spin-orbit CNOT gate, and four single-qubit operations performed with prisms and polarization components. Our results describe the implementation of arbitrary single-qubit channels on the photon polarization using the transverse mode as the ancillary qubit.
Inspired by the Solovay-Kitaev decomposition for approximating unitary operations as a sequence of operations selected from a universal quantum computing gate set, we introduce a method for approximating any single-qubit channel using single-qubit gates and the controlled-not (cnot). Our approach uses the decomposition of the single-qubit channel into a convex combination of "quasiextreme" channels. Previous techniques for simulating general single-qubit channels would require as many as 20 cnot gates, whereas ours only needs one, bringing it within the range of current experiments.
The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of π/2kπ/ 2^kπ/2k, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and π/8π/ 8π/8 gates. The algorithm runs in O(log⁡2.71(1/ε))O(\log^{2.71}(1/ε))O(log2.71(1/ε)) time, and produces as output a sequence of O(log⁡3.97(1/ε))O(\log^{3.97}(1/ε))O(log3.97(1/ε)) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within ε>0ε> 0ε>0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d)SU(d)SU(d).
A quantum channel describes the general evolution of quantum systems, which is therefore a critical issue for quantum simulation and a fundamental element of quantum information. To realize quantum channels, an efficient method is to randomly implement Kraus operators, which does not require any ancillary quantum system. For open systems, it is natural to extend the method from implementation of unitary Kraus operators to nonunitary ones. However, when some Kraus operators are not proportional to unitary ones, this method becomes probabilistic even if the channel is trace preserving. In this paper, to overcome this drawback, we propose an algorithm to deterministically realize arbitrary trace-preserving channels with only one ancillary qubit and finite iterations of evolutions of the combined states of the system and the ancilla. Moreover, to show the validity of the method, we experimentally realize conventional and modified Landau-Streater channels based on the algorithm. Our results shed light on quantum simulation of quantum channels for open systems.
Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of ε>0ε> 0ε>0 requires O(log⁡c[1/ε])\text{O}(\log^c[1/ε])O(logc[1/ε]) quantum gates with c>1.44042c > 1.44042c>1.44042. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using Ω(log⁡[1/ε])Ω(\log[1/ε])Ω(log[1/ε]) gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using O(log⁡[1/ε]log⁡log⁡[1/ε]log⁡log⁡log⁡[1/ε]⋯ )\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)O(log[1/ε]loglog[1/ε]logloglog[1/ε]⋯) FT gates chosen from a finite set depending on the value of εεε. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy ε>0ε> 0ε>0. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.
The Depolarizing channel is one of the most important noise models and constitutes a reliable benchmark for the quantum information field. In this work, we present a simple way to emulate a Depolarizing channel by exploring a maximally non separable spin-orbit mode in a compact linear optical circuit. The evolution of different states has been successfully reproduced. Our results are in excellent agreement compared with the results obtained by the spin-orbit Solovay-Kitaev decomposition for the Depolarizing channel, also presented here for the first time. Our proposal can be a powerful tool for studies of depolarizing effect.