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arXiv:1003.5760 · Physical Review A

Quantum-state preparation with universal gate decompositions

Martin Plesch, Časlav Brukner

Published Mar 3, 2011• 335 citations
arXivHTMLPDF

Abstract

In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of $2$. It also reduces the total number of cnot gates from ${2}^{n}$ to $\frac{23}{24}{2}^{n}$ in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from $11$ cnot gates to $9$ and the depth from $11$ to $5$ steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.

DOI
10.1103/physreva.83.032302

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We consider the decomposition of arbitrary isometries into a sequence of single-qubit and controlled-not (cnot) gates. In many experimental architectures, the cnot gate is relatively costly and hence we aim to keep the number of these as low as possible. We derive a theoretical lower bound on the number of cnot gates required to decompose an arbitrary isometry from mmm to nnn qubits and give three explicit gate decompositions that achieve this bound up to a factor of about 2 in the leading order. We also perform some further optimizations for certain cases where mmm and nnn are small. In addition, we show how to apply our result for isometries to give a decomposition scheme for an arbitrary quantum operation via Stinespring's theorem and derive a lower bound on the number of cnot gates in this case too. These results will have an impact on experimental efforts to build a quantum computer, enabling them to go further with the same resources.
Multi-controlled unitary gates have been a subject of interest in quantum computing since its inception, and are widely used in quantum algorithms. The current state-of-the-art approach to implementing n-qubit multi-controlled gates involves the use of a quadratic number of single-qubit and CNOT gates. However, linear solutions are possible for the case where the controlled gate is a special unitary SU(2). The most widely-used decomposition of an n-qubit multi-controlled SU(2) gate requires a circuit with a number of CNOT gates proportional to 28n. In this work, we present a new decomposition of n-qubit multi-controlled SU(2) gates that requires a circuit with a number of CNOT gates proportional to 20n, and proportional to 16n if the SU(2) gate has at least one real-valued diagonal. This new approach significantly improves the existing algorithm by reducing the number of CNOT gates and the overall circuit depth. As an application, we show the use of this decomposition for sparse quantum state preparation. Our results are further validated by demonstrating a proof of principle on a quantum device accessed through quantum cloud services.
We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1∣21|21∣2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.
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Universal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to π rotations up to a global phase. We show that a gate set comprised of π rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two π rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled π rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.