One of the main goals in quantum circuit optimisation is to reduce the number of ancillary qubits and the depth of computation, to obtain robust computation. However, most of known techniques, based on local rewriting rules, for parallelising quantum circuits will require the addition of ancilla qubits, leading to an undesired space-time tradeoff. Recently several novel approaches based on measurement-based quantum computation (MBQC) techniques attempted to resolve this problem. The key element is to explore the global structure of a given circuit, defined via translation into a corresponding MBQC pattern. It is known that the parallel power of MBQC is superior to the quantum circuit model, and hence in these approaches one could apply the MBQC depth optimisation techniques to achieve a lower depth. However, currently, once the obtained parallel pattern is translated back to a quantum circuit, one should either increase the depth or add ancilla qubits. In this paper we characterise those computations where both optimisation could be achieved together. In doing so we present a new connection between two MBQC depth optimisation procedures, known as the maximally delayed generalised flow and signal shifting. This structural link will allow us to apply an MBQC qubit optimisation procedure known as compactification to a large class of pattern including all those obtained from any arbitrary quantum circuit. We also present a more efficient algorithm (compared to the existing one) for finding the maximally delayed generalised flow for graph states with flow.
Measurement-based quantum computing (MBQC), an alternate paradigm for formulating quantum algorithms, can lead to potentially more flexible and efficient implementations as well as to theoretical insights on the role of entanglement in a quantum algorithm. Using the graph-theoretical ZX-calculus, we describe and apply a general scheme for reformulating quantum circuits as MBQC implementations. After illustrating the method using the two-qubit Deutsch–Jozsa algorithm, we derive a ZX graph-diagram that encodes a general MBQC implementation for the three-qubit Deutsch–Jozsa algorithm. This graph describes an 11-qubit cluster state on which single-qubit measurements are used to execute the algorithm. Particular sets of choices of the axes for the measurements can be used to implement any realization of the oracle. In addition, we derive an equivalent lattice cluster state for the algorithm.
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.
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.
Diagrammatic reasoning via languages such as ZX Calculus is a powerful tool in theoretical quantum computing. However, physical quantum computers do not understand diagrams and require explicit quantum circuits as inputs. Finding a circuit equivalent to the given diagram is known as circuit extraction. Such problems in graphical calculi form a limitation preventing the wider adoption of diagrammatic reasoning. Computational complexity theory formalises the difficulty of problems. For ZX diagrams, the circuit extraction problem is #P-hard and unlikely to have an efficient algorithm. Therefore, partial solutions are critical. Currently, efficient circuit extraction is only possible for diagrams containing a structure of so-called Pauli flow. This structure is necessary and sufficient for robust determinism in measurement-based quantum computation (MBQC) – a model of quantum computation, an alternative to circuits, where adaptive single-qubit measurements drive computation. Thus, Pauli flow links MBQC, ZX Calculus, and the circuit model. It is known how to find flow structures in polynomial time when they exist; nevertheless, their lengthy and complex definitions often hinder working with them. In this thesis, I present contributions to two connected topics: the hardness of problems in graphical calculi and Pauli flow as a method for overcoming circuit extraction. First, I show that circuit extraction can be #P-hard for phase-free ZH diagrams, extending the previous result for ZX diagrams. I also establish a relation between phase-free ZH and
NP⧣P class. This involves weakening the oracle, crafting a complete problem, and encoding the problem into phase-free ZH. My next contribution concerns Pauli flow. I simplify the Pauli flow definition by providing a new algebraic interpretation. This involves defining two matrices arising from the adjacency matrix of the underlying graph. Then, Pauli flow corresponds to the existence of a right-inverse of the first matrix, which, when multiplied by the second matrix, results in a matrix of a directed acyclic graph. Based on the newly defined algebraic interpretation, I obtain O(n
3) algorithms for finding Pauli flow, improving on the previous O(n
4) bounds. I also introduce a lower bound for Pauli flow-finding by linking it to linear algebra problems over F
2. Finally, when given an unlabelled open graph, I show how to find measurement labelling resulting in Pauli flow.
Abstract The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions—a so-called standard form—which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus, enabling a significant decrease in circuit depth for certain applications. We show that groups of mutually-commuting operators can be implemented using fully-parallel, i.e. non-adaptive, measurements within our protocol. In addition, we discuss how groups of mutually commuting observables can be simulatenously measured by adjusting the resource state, rather than performing a costly basis transformation prior to the measurement as it is done in the circuit model. Finally, we demonstrate the utility of our technique on two examples of high practical relevance—the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver (VQE) for the ground-state energy estimation of the water molecule. For the VQE, we find a reduction of the depth by a factor of 4 to 5 using measurement patterns vs. the standard circuit model. At the same time, since we incorporate the simultaneous measurements, our patterns allow us to save shots by a factor of at least 3.5 compared to measuring Pauli strings individually in the circuit model.