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arXiv:2104.00572 · Quantum

Flow conditions for continuous variable measurement-based quantum computing

Robert I. Booth, Damian Markham

Published Oct 19, 2023• 3 citations
arXivHTMLPDF

Abstract

In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement results. We introduce flow-based methods for quantum computation with continuous-variable graph states, which we call CV-flow. These are inspired by, but not equivalent to, the notions of causal flow and g-flow for qubit MBQC. We also show that an MBQC with CV-flow approximates a unitary arbitrarily well in the infinite-squeezing limit, addressing issues of convergence which are unavoidable in the infinite-dimensional setting. In developing our proofs, we provide a method for converting a CV-MBQC computation into a circuit form, analogous to the circuit extraction method of Miyazaki et al, and an efficient algorithm for finding CV-flow when it exists based on the qubit version by Mhalla and Perdrix. Our results and techniques naturally extend to the cases of MBQC for quantum computation with qudits of prime local dimension.

DOI
10.22331/q-2023-10-19-1146

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In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement results. We introduce flow-based methods for quantum computation with continuous-variable graph states, which we call CV-flow. These are inspired by, but not equivalent to, the notions of causal flow and g-flow for qubit MBQC. We also show that an MBQC with CV-flow approximates a unitary arbitrarily well in the infinite-squeezing limit, addressing issues of convergence which are unavoidable in the infinite-dimensional setting. In developing our proofs, we provide a method for converting a CV-MBQC computation into a circuit form, analogous to the circuit extraction method of Miyazaki et al, and an efficient algorithm for finding CV-flow when it exists based on the qubit version by Mhalla and Perdrix. Our results and techniques naturally extend to the cases of MBQC for quantum computation with qudits of prime local dimension.
Abstract In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement outcomes. We introduce flow-based methods for MBQC with qudit graph states, which we call Z d -flow, when the local dimension is an odd prime. Our main results are a proof that Z d -flow is a necessary and sufficient condition for a strong form of outcome determinism. Along the way, we find a suitable generalisation of the concept of measurement planes to this setting and characterise the allowed measurements in a qudit MBQC. We also provide a polynomial-time algorithm for finding an optimal Z d -flow whenever one exists.
Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits. Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an O(n3)O(n^3)O(n3) flow-finding algorithm (where nnn is the number of qudits), matching the best known complexity for qubit flows and improving on the previous O(n4)O(n^4)O(n4) result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow. Lastly, we propose an algorithmic approach to generating large qudit computations with flow, for testing or machine learning.
Abstract Measurement-based quantum computation (MBQC) is a protocol for quantum computation that represents a model distinct from the circuit-based approach. MBQC has been proposed not only for qubits but also for qudits, continuous-variable (CV) qubits, and Bose–Einstein condensates (BECs) qubits. In qubit-based MBQC, arbitrary rotations on the Bloch sphere can be performed by measuring a graph state. This naturally raises the question of whether arbitrary rotations on the Bloch sphere can similarly be achieved through measurements in other types of quantum bits. We have demonstrated that this can indeed be realized for BECs qubits by considering composite graph states involving CV qubits and BECs qubits.
We propose a new scheme of measurement-based quantum computation (MBQC) using an error-correcting code against photon-loss in circuit quantum electrodynamics. We describe a specific protocol of logical single-qubit gates given by sequential cavity measurements for logical MBQC and a generalised Schrödinger cat state is used for a continuous-variable (CV) logical qubit captured in a microwave cavity. To apply an error-correcting scheme on the logical qubit, we utilise a d-dimensional quantum system called a qudit. It is assumed that a three CV-qudit entangled state is initially prepared in three jointed cavities and the microwave qudit states are individually controlled, operated, and measured through a readout resonator coupled with an ancillary superconducting qubit. We then examine a practical approach of how to create the CV-qudit cluster state via a cross-Kerr interaction induced by intermediary superconducting qubits between neighbouring cavities under the Jaynes-Cummings Hamiltonian. This approach could be scalable for building 2D logical cluster states and therefore will pave a new pathway of logical MBQC in superconducting circuits toward fault-tolerant quantum computing.
Quantum computers, leveraging the principles of quantum mechanics, hold the potential to surpass classical computers in numerous applications, with implications across various domains. Besides the well-known gate model, Measurement-based Quantum Computation (MBQC) is another promising computational approach to achieve universal quantum computation. In MBQC, large ensembles of qubits are prepared in a highly entangled cluster state, forming the basis for executing quantum computations through sequential measurements. Cluster states are realized using both continuous variables (CV) and discrete variables (DV) techniques. In the CV-based methods, Frequency Domain Multiplexing (FDM), Time Domain Multiplexing (TDM), Spatial Domain Multiplexing (SDM), and hybrid schemes are employed. This paper thoroughly discusses and compares these approaches, elucidating their strengths and limitations. Additionally, the generation of photonic cluster states in DV is explored and some recent results are reported. Some recent state-of-the-art advancements in photonic and superconducting qubits entanglement, which can potentially serve as cluster states, are also presented. Finally, we highlight the approach that exhibits the most promising characteristics for achieving efficient cluster state realization in the context of MBQC.
We introduce a new characterisation of determinism in Measurement-Based Quantum Computing (MBQC). The one-way model consists in performing local measurements over a large entangled state represented by a graph. The ability to perform an overall deterministic computation requires a correction strategy because of the non-determinism of each measurement. The existence of such a correction strategy depends on the underlying open graph, which is a description of the resource state together with the basis of the performed measurements. GFlow is a well-known graphical characterisation of robust determinism in MBQC when every measurement is performed in some specific planes of the Bloch sphere. While Pauli measurements are ubiquitous in MBQC, GFlow fails to be necessary for determinism when a measurement-based quantum computation involves Pauli measurements. Pauli Flow was designed as a generalisation of GFlow to handle MBQC with Pauli measurements, and guarantees robust determinism, however, it has been shown more recently that it fails to be a necessary condition. Our contribution is twofold. First, we demonstrate that Pauli flow is actually necessary for robust determinism in a weaker sense: given an open graph, i.e. a resource state, a deterministic computation can be driven iff it has a Pauli flow. However, the Pauli flows do not reflect all the possible correction strategies over a particular resource state, and properties like measurement order or computational depth are not necessarily reflected by a Pauli flow. Thus, to characterise determinism in full generality, we introduce a further extension called Shadow Pauli Flow that we prove necessary and sufficient for robust determinism: An MBQC is robustly deterministic if and only if its correction strategy is consistent with a Shadow Pauli flow. Furthermore, we show that Shadow Pauli flow can be computed in polynomial time.