results for au:Bocharov_A in:quant-ph

- Jun 09 2016 quant-ph arXiv:1606.02315v2Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate $R=|0\rangle \langle 0| + |1\rangle \langle 1| - |2\rangle \langle 2|$. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any two-level Householder reflection to precision $\varepsilon$ by a metaplectic circuit of $R$-count at most $C \, \log_3(1/\varepsilon) + O(\log \log 1/\varepsilon)$ with $C=8$. A new result in this note takes the constant down to $C=5$ for non-exceptional target reflections under a certain credible number-theoretical conjecture. The new method increases the chances of obtaining a truly optimal circuit but may not guarantee the true optimality. Efficient approximations of an important ternary quantum gate proposed by Howard, Campbell and others is also discussed. Apart from this, the note is mostly didactical: we demonstrate how to leverage Lenstra's integer geometry algorithm from 1983 for circuit synthesis.
- We determine the cost of performing Shor's algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to emulate the standard binary version of the algorithm by encoding each qubit in a three-level system. We compare the two approaches and analyze the complexity of implementing Shor's period finding function in the two models. We also highlight the fact that the cost of achieving universality through magic states in MTQC architecture is asymptotically lower than in generic ternary case.
- Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only $O(\log\,n)$, while using with $O(n)$ ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate $C(X): |i,j\rangle \mapsto |i, j + \delta_{i,2} \mod 3\rangle$ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 \pi \, i/9},1,e^{2 \pi \, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.
- Oct 15 2015 quant-ph arXiv:1510.03888v1We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+$\pi/12$ [arXiv:1409.3552], running on average in time polynomial in $O(\log(1/\varepsilon))$ (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer.
- Apr 15 2015 quant-ph arXiv:1504.03383v3A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary $n$-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in $O(3^{2\,n} \, \log{1/\varepsilon})$ and entanglement cost that is exponential in $n$. Another flavor of the algorithm produces efficient circuits with upper complexity bound in $O(n\,3^{2\,n} \, \log{1/\varepsilon})$ and no additional entanglement cost.
- Dec 19 2014 quant-ph arXiv:1412.5608v4We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+$T$ basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit. The equivalent $T$-count of the synthesized circuit is bounded by $k \, C_0 \log_2(1/\varepsilon) + E(n,k)$, where $k$ is the number of distinct phases in the diagonal $n$-qubit unitary, $\varepsilon$ is the desired precision, $C_0$ is a quality factor of the implementation method ($1<C_0<4$), and $E(n,k)$ is the total entanglement cost (in $T$ gates). We determine an optimal decision boundary in $(k,n,\varepsilon)$-space where our decomposition algorithm achieves lower entanglement cost than previous state-of-the-art techniques. Our method outperforms state-of-the-art techniques for a practical range of $\varepsilon$ values and diagonal operators and can reduce the number of $T$ gates exponentially in $n$ when $k << 2^n$.
- Dec 03 2014 quant-ph arXiv:1412.1033v1Recently Neil Ross and Peter Selinger analyzed the problem of approximating z- rotations by means of single-qubit Clifford+T circuits. Their main contribution is a deterministic-search technique which allowed them to make approximating circuits shallower. We adapt the deterministic-search technique to the case of Pauli+V circuits and prove similar results. Because of the relative simplicity of the Pauli+V framework, we use much simpler geometric methods.
- Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of $T$ gates of about $1/3$ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+$T$ 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+$T$ 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 multi-qubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols terminate after a finite number of steps. Specifically, we apply our method to the Clifford+$T$, Clifford+$V$, and Clifford+$\pi/12$ gate sets to achieve decompositions with expected gate counts of $\log_b(1/\varepsilon)+O(\log(\log(1/\varepsilon)))$, where $b$ is a quantity related to the expansion property of the underlying universal gate set.
- Apr 23 2014 quant-ph arXiv:1404.5320v2Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a $2.5$ times reduction in expected $T$-count over ancilla-free techniques for single-qubit unitary decomposition. However, the previously best known algorithm to synthesize RUS circuits requires exponential classical runtime. In this paper we present an algorithm to synthesize an RUS circuit to approximate any given single-qubit unitary within precision $\varepsilon$ in probabilistically polynomial classical runtime. Our synthesis approach uses the Clifford+$T$ basis, plus one ancilla qubit and measurement. We provide numerical evidence that our RUS circuits have an expected $T$-count on average $2.5$ times lower than the theoretical lower bound of $3 \log_2 (1/\varepsilon)$ for ancilla-free single-qubit circuit decomposition.
- In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations for non-abelian quasiparticles described by the Fibonacci anyon model. We develop a probabilistically polynomial algorithm that outputs a braid pattern to approximate a given single-qubit unitary to a desired precision. We also classify the single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern and present an efficient algorithm to produce their braid patterns. Our techniques produce braid patterns that meet the uniform asymptotic lower bound on the compiled circuit depth and thus are depth-optimal asymptotically. Our compiled circuits are significantly shorter than those output by prior state-of-the-art methods, resulting in improvements in depth by factors ranging from 20 to 1000 for precisions ranging between $10^{-10}$ and $10^{-30}$.
- Mar 07 2013 quant-ph arXiv:1303.1411v1We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis. The $V$ basis is an alternative universal basis to the more commonly studied $\{H,T\}$ basis. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in precision $\log(1/\epsilon)$) and offers a depth/precision guarantee that improves upon state-of-the-art methods for compiling into the $\{H,T\}$ basis by factors ranging from 1.86 to $\log_2(5)$. The second algorithm is analogous to direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, and requires time exponential in $\log(1/\epsilon)$; however, we show that in practice the runtime is reasonable for an important range of target precisions.
- Jun 15 2012 quant-ph arXiv:1206.3223v1Given an arbitrary single-qubit operation, an important task is to efficiently decompose this operation into an (exact or approximate) sequence of fault-tolerant quantum operations. We derive a depth-optimal canonical form for single-qubit quantum circuits, and the corresponding rules for exactly reducing an arbitrary single-qubit circuit to this canonical form. We focus on the single-qubit universal H,T basis due to its role in fault-tolerant quantum computing, and show how our formalism might be extended to other universal bases. We then extend our canonical representation to the family of Solovay-Kitaev decomposition algorithms, in order to find an \epsilon-approximation to the single-qubit circuit in polylogarithmic time. For a given single-qubit operation, we find significantly lower-depth \epsilon-approximation circuits than previous state-of-the-art implementations. In addition, the implementation of our algorithm requires significantly fewer resources, in terms of computation memory, than previous approaches.