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arXiv:1209.5079 · Physical Review A

Compact quantum circuits from one-way quantum computation

Raphael Dias da Silva, Ernesto F. Galvão

Published Jul 19, 2013• 9 citations
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Abstract

In this paper we address the problem of translating one-way quantum computation (1WQC) into the circuit model. We start by giving a straightforward circuit representation of any 1WQC, at the cost of introducing many ancilla wires. We then propose a set of simple circuit identities that explore the relationship between the entanglement resource and correction structure of a 1WQC, allowing one to obtain equivalent circuits acting on fewer qubits. We conclude with some examples and a discussion of open problems.

DOI
10.1103/physreva.88.012319

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In this paper we address the problem of translating one-way quantum computation (1WQC) into the circuit model. We start by giving a straightforward circuit representation of any 1WQC, at the cost of introducing many ancilla wires. We then propose a set of simple circuit identities that explore the relationship between the entanglement resource and correction structure of a 1WQC, allowing one to obtain equivalent circuits acting on fewer qubits. We conclude with some examples and a discussion of open problems.
In one-way quantum computations (1WQC), quantum correlations in an entangled state, called a cluster or graph state, are exploited to perform universal quantum computations using single-qubit measurements. The choices of bases for these measurements as well as the structure of the entanglements specify a quantum algorithm. The needed computations in this model are organized as patterns. Previously, an approach to extract a 1WQC pattern from a quantum circuit was proposed. The approach takes a quantum circuit consisting of only CZ andj(oc) gates and translates it into a 1WQC pattern after performing some known optimizations. However, most quantum synthesis algorithms decompose circuits using a library containing CNOT and any single-qubit gates. In this paper, we show how this approach can be modified in a way that it can take a circuit consisting of CNOT and any single-qubit gates as well as CZ and J(oc) gates to produce a 1WQC pattern. Experimental results show that adding CNOT gate directly to the pattern decreases the translation run-time.
We present a translation from the standard network model of quantum computation to the one-way model of quantum computation. The translation is compositional, i.e., it preserves the structure of computations, which allows us to abstract the concrete realizations in a monadic abstraction layer. We briefly review the By-Product and Measurement Calculus approaches to combining circuits in the one-way model, and show how the procedures can be expressed in the exact same notation, bearing a direct relationship with the representation of circuits in the standard network model, and using monads. Discussions on improving the abstraction yield us to introducing an alternative approach to combining circuits in the one-way model: a graphical one.
One-way quantum computation (1WQC) is a model of universal quantum computations in which a specific highly entangled state called acluster stateallows for quantum computation by single-qubit measurements. The needed computations in this model are organized as measurement patterns. The traditional approach to obtain a measurement pattern is by translating a quantum circuit that solely consists of CZ andJ(α) gates into the corresponding measurement patterns and then performing some optimizations by using techniques proposed for the 1WQC model. However, in these cases, the input of the problem is a quantum circuit, not an arbitrary unitary matrix. Therefore, in this article, we focus on the first phase—that is, decomposing a unitary matrix into CZ andJ(α) gates. Two well-known quantum circuit synthesis methods, namely cosine-sine decomposition and quantum Shannon decomposition are considered and then adapted for a library of gates containing CZ andJ(α), equipped with optimizations. By exploring the solution space of the combinations of these two methods in a bottom-up approach of dynamic programming, a multiobjective quantum circuit synthesis method is proposed that generates a set of quantum circuits. This approach attempts to simultaneously improve the measurement pattern cost metrics after the translation from this set of quantum circuits.
One-way measurement based quantum computations (1WQC) may describe unitary transformations, via a composition of CPTP maps which are not all unitary themselves. This motivates the following decision problems. Is it possible to determine whether a "quantum-to-quantum" 1WQC procedure (having non-trivial input and output subsystems) performs a unitary transformation? Is it possible to describe precisely how such computations transform quantum states, by translation to a quantum circuit of comparable complexity? In this article, we present an efficient algorithm for transforming certain families of measurement-based computations into a reasonable unitary circuit model, in particular without employing the principle of deferred measurement.
Quantum computing allows the realization of certain tasks more efficiently than its classical counterpart. This monograph seeks to present a type of quantum computation that is different from the standard form: Measurement-Based Quantum Computation, more specifically a model known as One-Way Quantum Computation (1WQC), developed by R. Raussendorf and H. Briegel. To understand it, this work starts from the circuit model, the most common type of quantum computing, describing its main components. The 1WQC model is then introduced, with an attempt to didactically present its operation. A way of applying one- and two-qubit quantum logic gates is also shown, thus explaining how universal quantum computing is possible in this model. As an example, the translation of Deutsch’s algorithm to this model is also discussed. The result of this operation raises, however, open questions concerning the translation of Oracle-type algorithms, such as Deutsch’s, to the 1WQC model
In one-way quantum computation (1WQC) model, universal quantum computations are performed using measurements to designated qubits in a highly entangled state. The choices of basis for these measurements as well as the structure of the entanglements specify a quantum algorithm. Although a number of methods have been proposed to simulate quantum circuit model on classical computers, no efficient tool has been developed to simulate the 1WQC model directly. In this paper, some techniques such as qubit elimination, implicit and in-place matrix-vector multiplication and pattern reordering are utilized to considerably reduce the time and memory needed for the simulations. These techniques were implemented in a tool called One-Way Quantum computation Simulator (OWQS). Experimental results validate the efficiency of the proposed approach.