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Quantum Circuits: From a Network to a One-Way Model - A Tutorial Version

Larisse D. Voufo, Gerardo Ortíz, Amr Sabry

Published Jan 1, 2007• 1 citation
OpenAlex

Abstract

We present elements of quantum circuits trans- lations from the (standard) network or circuit model to the one-way one. We present a translation scheme, give an account of currently existing tools to apply the scheme, and propose an extension of those tools into a complete translation calculus. We analyse the set of difficulties incurred from such work, and show an engendered opening to new sets of problems and ideas. Among others, this paper introduces the notions of graphical concatenation, graph state reduction (GSR) and graph state extension (GSE) passes, as well as an algorithm for running the (extended) measurement calculus with acceptable efficiency.

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We present elements of quantum circuits trans- lations from the (standard) network or circuit model to the one-way one. We present a translation scheme, give an account of currently existing tools to apply the scheme, and propose an extension of those tools into a complete translation calculus. We analyse the set of difficulties incurred from such work, and show an engendered opening to new sets of problems and ideas. Among others, this paper introduces the notions of graphical concatenation, graph state reduction (GSR) and graph state extension (GSE) passes, as well as an algorithm for running the (extended) measurement calculus with acceptable efficiency.
Complete sets of graphical rewrite rules enable fully graphical reasoning about quantum computations and have been an area of active research for more than a decade. Many recent applications of the ZX-calculus have made use of the close correspondence between ZX-diagrams and computations in the one-way model of measurement-based quantum computation. In this model, various kinds of flow properties ensure deterministic implementability; for ZX-diagrams, these same properties allow efficient translation into quantum circuits (a problem that is known to be #P-hard in general). Therefore, flow-preserving ZX-calculus rewrite rules are of strong interest. Here, we extend the set of flow-preserving rules appearing in the literature with a few new rules and extensions of existing rules. We then show that the resulting rule set is complete for all flow-preserving translations between ZX-diagrams of appropriate form. The proof employs a manifestly flow-preserving equivalent of circuit extraction, where a diagram with gflow is transformed, using only flow-preserving rewrite rules, into a diagram with causal flow.
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.
The one-way model of Measurement-Based Quantum Computing and the gate-based circuit model give two different presentations of how quantum computation can be performed. There are known methods for converting any gate-based quantum circuit into a one-way computation, whereas the reverse is only efficient given some constraints on the structure of the measurement pattern. Causal flow and generalised flow have already been shown as sufficient, with efficient algorithms for identifying these properties and performing the circuit extraction. Pauli flow is a weaker set of conditions that extends generalised flow to use the knowledge that some vertices are measured in a Pauli basis. In this paper, we show that Pauli flow can similarly be identified efficiently and that any measurement pattern whose underlying graph admits a Pauli flow can be efficiently transformed into a gate-based circuit without using ancilla qubits. We then use this relationship to derive simulation results for the effects of graph-theoretic rewrites in the ZX-calculus using a more circuit-like data structure we call the Pauli Dependency DAG.
The one-way model of measurement-based quantum computing implements computations via successive adaptive single-qubit measurements on a resource graph state. This model has practical applications, particularly in photonics, and it is also useful as a theoretical tool e.g. for optimisation. Gflow is a necessary and sufficient condition for implementing certain one-way computations deterministically (in a suitable sense); it is also used in efficient translations from the one-way model to quantum circuits. For a computation on nnn qubits, a gflow can be found in O(n3)\mathcal{O}(n^3)O(n3) time. Here, we consider an incompletely specified computation given by an unlabelled open graph: the graph state as well as the input and output qubits are known, but the measurements have not yet been fixed. We give an algorithm that identifies a measurement labelling and a compatible gflow, and runs in O(n3)\mathcal{O}(n^3)O(n3), strictly generalising the previous approach. The new algorithm can also handle restrictions on the order of the measurements and returns only solutions compatible with these constraints. We additionally prove that if an open graph has equal numbers of inputs and outputs, it has at most one labelling compatible with gflow; and show how to identify additional inputs for an open graph that does not yet have the maximal number, without breaking an existing gflow.
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.
Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of 333-edge-colorings of 333-regular planar graphs.