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.
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.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of unitary transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to "unitary circuit models", which are models of computation in which unitary operators are expressed as a product of "elementary" unitary transformations. However, unitary transformations can also be effected as a composition of operations which are not all unitary themselves: the "one-way measurement model" is one such model of quantum computation. In this thesis, we examine the relationship between representations of unitary operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement model can be described as simulating a unitary circuit, by considering the combinatorial structures which are common to unitary circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from unitary transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the unitary circuit model.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized of probability, and computations are most commonly described as products of transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to unitary circuit models, which are models of computation in which operators are expressed as a product of elementary transformations. However, transformations can also be effected as a composition of operations which are not all themselves: the measurement model is one such of quantum computation.
In this thesis, we examine the relationship between representations of operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement can be described as simulating a circuit, by considering the combinatorial structures which are common to circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the circuit model.
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 the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the so-called cluster model or one-way quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms.
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.
Equivalence checking of hybrid quantum circuits is of primary importance, given that quantum circuit transformations are omnipresent along the quantum compiler chain. While some approaches exist for automating this task, most focus on the simple case of unitary circuits. At the same time, real quantum computing requires hybrid circuits equipped with measurement operators. Moreover, the few approaches targeting the hybrid case are limited to a restricted class of problems. We propose tackling the Quantum Hybrid Circuit Equivalence Checking problem through lifting unitary circuit verification using a transformation known as deferred measurement. We show that this approach alone significantly outperforms prior work, and that, with the addition of specific unitary-level techniques we call separation and projection, it can handle much larger classes of hybrid circuit equivalence problems. We have implemented and evaluated our method over standard circuit transformations such as teleportation, one-way measurement, or the IBM Qiskit compiler, demonstrating its promises. As a side finding, we have identified and reported several unexpected behaviours with the Qiskit compiler.