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arXiv:1311.1074

Repeat-Until-Success: Non-deterministic decomposition of single-qubit unitaries

Adam Paetznick, Krysta M. Svore

Published Nov 5, 2013• 20 citations
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Abstract

We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance $ε$ and requires significantly fewer non-Clifford gates than existing techniques. We develop "Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit $Z$-axis rotation to within distance $ε$, where the number of $T$ gates scales as $1.26\log_2(1/ε) - 3.53$, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of $2.4\log_2(1/ε) - 3.28$ can be achieved for arbitrary unitaries and a small range of $ε$, which is roughly twice as good as optimal deterministic decomposition methods.

DOI
10.48550/arxiv.1311.1074

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Vadym Kliuchnikov, Dmitri A Maslov, Michele Mosca

Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision $\ensuremath{\epsilon}$ requires $\ensuremath{\Omega}\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires $O\mathbf{(}{log}^{3+\ensuremath{\delta}}(1/\ensuremath{\epsilon})\mathbf{)}$ gates and does not use ancillae and the phase kickback approach that requires $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates but uses $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathbf{)}$ ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision $\ensuremath{\epsilon}$ using $O\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ Clifford and $T$ gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagrange's four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ operations with integers.

arXiv:1212.0822•Physical Review Letters•May 8, 2013•127 citations
We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance εεε and requires significantly fewer non-Clifford gates than existing techniques. We develop "Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit ZZZ-axis rotation to within distance εεε, where the number of TTT gates scales as 1.26log⁡2(1/ε)−3.531.26\log_2(1/ε) - 3.531.26log2​(1/ε)−3.53, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4log⁡2(1/ε)−3.282.4\log_2(1/ε) - 3.282.4log2​(1/ε)−3.28 can be achieved for arbitrary unitaries and a small range of εεε, which is roughly twice as good as optimal deterministic decomposition methods.
We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance ϵ\epsilonϵ and requires significantly fewer non-Clifford gates than existing techniques. We develop ``Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit ZZZ-axis rotation to within distance ϵ\epsilonϵ, where the number of TTT gates scales as 1.26log⁡2(1/ϵ)−3.531.26\log_2(1/\epsilon) - 3.531.26log2​(1/ϵ)−3.53, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4log⁡2(1/ϵ)−3.282.4\log_2(1/\epsilon) - 3.282.4log2​(1/ϵ)−3.28 can be achieved for arbitrary unitaries and a small range of ϵ\epsilonϵ, which is roughly twice as good as optimal deterministic decomposition methods.
We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.
Recently it was shown that the resources required to implement unitary operations on a quantum computer can be reduced by using probabilistic quantum circuits called repeat-until-success (RUS) circuits. However, the previously best-known algorithm to synthesize a RUS circuit for a given target unitary requires exponential classical runtime. We present a probabilistically polynomial-time algorithm to synthesize a RUS circuit to approximate any given single-qubit unitary to precision ϵ over the Clifford+T basis. Surprisingly, the T count of the synthesized RUS circuit surpasses the theoretical lower bound of 3 log_{2}(1/ϵ) that holds for purely unitary single-qubit circuit decomposition. By taking advantage of measurement and an ancilla qubit, RUS circuits achieve an expected T count of 1.15 log_{2}(1/ϵ) for single-qubit z rotations. Our method leverages the fact that the set of unitaries implementable by RUS protocols has a higher density in the space of all unitaries compared to the density of purely unitary implementations.
Repeat-until-success (RUS) circuits can approximate a given single-qubit unitary with an expected number of TTT gates of about 13\frac{1}{3}31​ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford + TTT gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford + TTT gate set. The protocol, which we call probabilistic quantum circuits with fallback (PQF), implements a walk on a discrete Markov chain in which the target unitary is an absorbing state and in which transitions are induced by multiqubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols are guaranteed to terminate after a finite number of steps. Specifically, we apply our method to the Clifford + TTT, Clifford + VVV, and Clifford + $\ensuremath{\pi}/12$ gate sets to achieve decompositions with expected gate counts of ${log}_{b}(1/\ensuremath{\varepsilon})+O{ln[ln(1/\ensuremath{\varepsilon})]}$, where bbb is a quantity related to the expansion property of the underlying universal gate set.
Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision $\ensuremath{\epsilon}$ requires $\ensuremath{\Omega}\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires $O\mathbf{(}{log}^{3+\ensuremath{\delta}}(1/\ensuremath{\epsilon})\mathbf{)}$ gates and does not use ancillae and the phase kickback approach that requires $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates but uses $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathbf{)}$ ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision $\ensuremath{\epsilon}$ using $O\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ Clifford and TTT gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagrange's four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ operations with integers.