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arXiv:1506.04394 · Physical Review A

Optimal synthesis of multivalued quantum circuits

Yao-Min Di, Hai‐Rui Wei

Published Dec 9, 2015• 24 citations
arXivHTMLPDF

Abstract

Although much work has been devoted to multivalued quantum logic synthesis, the question of whether multivalued quantum circuits are more efficient than conventional binary quantum circuits is still open. This article is devoted to the optimization of generic multivalued quantum circuits. Multivalued quantum Shannon decompositions are improved so that the circuits obtained are asymptotically optimal for all dimensionality $d$. The syntheses of uniformly multifold controlled ${R}_{y}$ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower bound of complexity for multivalued quantum circuits is investigated, and a quantity known as the efficiency index is proposed to evaluate the efficiency of the synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces improved results for all dimensionality $d$, whether the number of qudits $n$ is small or large. The two facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of $d\ensuremath{-}1$ in comparison to that for qubit circuits and the asymptotic efficiency index is increased with an increase of dimensionality $d$, reveal the potential advantage of qudit circuits over generic qubit circuits. Generic $n$-qudit circuits with $d\ensuremath{\ge}5$ and generic two-ququart circuits synthesized by the algorithm given here are practical circuits which are more efficient than the most efficient qubit circuits.

DOI
10.1103/physreva.92.062317

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Although much work has been devoted to multivalued quantum logic synthesis, the question of whether multivalued quantum circuits are more efficient than conventional binary quantum circuits is still open. This article is devoted to the optimization of generic multivalued quantum circuits. Multivalued quantum Shannon decompositions are improved so that the circuits obtained are asymptotically optimal for all dimensionality ddd. The syntheses of uniformly multifold controlled Ry{R}_{y}Ry​ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower bound of complexity for multivalued quantum circuits is investigated, and a quantity known as the efficiency index is proposed to evaluate the efficiency of the synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces improved results for all dimensionality ddd, whether the number of qudits nnn is small or large. The two facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of d\ensuremath−1d\ensuremath{-}1d\ensuremath−1 in comparison to that for qubit circuits and the asymptotic efficiency index is increased with an increase of dimensionality ddd, reveal the potential advantage of qudit circuits over generic qubit circuits. Generic nnn-qudit circuits with d\ensuremath≥5d\ensuremath{\ge}5d\ensuremath≥5 and generic two-ququart circuits synthesized by the algorithm given here are practical circuits which are more efficient than the most efficient qubit circuits.
Circuit synthesis of multi-controlled gates is crucial for qudit (ddd-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general (n−1)(n-1)(n−1)-controlled unitaries on nnn qudits, we reduce the controlled-increment (CINC) and generalized controlled-XXX (GCX) gate counts to O(n2)O(n^2)O(n2), improving upon existing O(n2+log⁡2d)O(n^{2+\log_2 d})O(n2+log2​d) CINC and O(n3)O(n^3)O(n3) GCX bounds. For (n−1)(n-1)(n−1)-controlled special unitaries, this complexity is further reduced to O(n)O(n)O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from nnn to mmm qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when ddd is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general nnn-qudit unitaries.
We propose the generalized controlled XXX (gcx) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Shannon decomposition (QSD), the most powerful technique for the synthesis of generic qubit circuits, to the qudit case. A comparison of ququart (d=4d=4d=4) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.
This paper proposes an optimized quantum block-ZXZZXZZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.
This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have 22484n−322n+53\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}4822​4n−23​2n+35​ CNOT gates, which is less that the best known exact decomposition algorithm by (4n−2−1)/3(4^{n-2} -1)/3(4n−2−1)/3 CNOT gates.
For practical usage of quantum systems in solving real-world problems, large amounts of classical data are required to be transferred/encoded to the quantum domain. Arbitrary classical data is usually encoded onto quantum devices by synthesizing and initializing a corresponding quantum state. Current techniques of arbitrary state synthesis, however, produce deep and complex quantum circuits, leading to low state fidelity and possible violations of decoherence constraints. In this work, we propose an improved methodology and optimized circuits for synthesizing any arbitrary quantum state from given classical data. Compared to existing methods, the proposed methodology results in circuits with lower gate count, lower circuit depth, and high state fidelity. The proposed methods are evaluated by simulation in MATLAB and IBM qasm, and realistic implementation on an IBM quantum device. The experimental results show a reduction in gate count and circuit depth by a factor of two over existing methods.
To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with O(k2m2/n)O(k^2 m^2/n)O(k2m2/n) quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is Ω(mϵ)\Omega(m^\epsilon)Ω(mϵ) (for any ϵ>0\epsilon > 0ϵ>0), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.