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arXiv:2312.17325 · Physical Review Letters

Nonunitary Gates Using Measurements Only

Daniel Azses, Jonathan Ruhman, Eran Sela

Published Dec 26, 2024• 0 citations
arXivHTMLPDF

Abstract

Measurement-based quantum computation (MBQC) is a universal platform to realize unitary gates, only using measurements that act on a preprepared entangled resource state. By deforming the measurement bases, as well as the geometry of the resource state, we show that MBQC circuits always transmit and act on the input state but generally realize nonunitary logical gates. In contrast to the stabilizer formalism that is often used for unitary gates, we find that ZX-calculus is an ideal computation method for these nonunitary gates. As opposed to unitary gates, nonunitary gates cannot be applied with certainty, due to the randomness of quantum measurements. We maximize the success probability of realizing nonunitary gates and discuss applications including imaginary time evolution, which we demonstrate on a noisy intermediate-scale quantum device.

DOI
10.1103/physrevlett.133.260603

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Measurement-based quantum computation (MBQC) is a universal platform to realize unitary gates, only using measurements that act on a preprepared entangled resource state. By deforming the measurement bases, as well as the geometry of the resource state, we show that MBQC circuits always transmit and act on the input state but generally realize nonunitary logical gates. In contrast to the stabilizer formalism that is often used for unitary gates, we find that ZX-calculus is an ideal computation method for these nonunitary gates. As opposed to unitary gates, nonunitary gates cannot be applied with certainty, due to the randomness of quantum measurements. We maximize the success probability of realizing nonunitary gates and discuss applications including imaginary time evolution, which we demonstrate on a noisy intermediate-scale quantum device.
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A quantum circuit is generalized to a nonunitary one whose constituents are nonunitary gates operated by quantum measurement. It is shown that a specific type of one-qubit nonunitary gates, the controlled-NOT gate, and all one-qubit unitary gates constitute a universal set of gates for the nonunitary quantum circuit, without the necessity of introducing ancilla qubits. A reversing measurement scheme is used to improve the probability of successful nonunitary gate operation. A quantum NAND gate and Abrams–Lloyd's nonlinear gate are analyzed as examples. Our nonunitary circuit can be used to reduce the qubit overhead needed to ensure fault-tolerant quantum computation.
Measurement based quantum computation (MBQC) is an effective paradigm for universal quantum computation. In this scheme, the universal set of quantum gates are realized by only local measurements on the prior prepared cluster states. The inevitable decoherence is harmful to the realization of those quantum gates. Here, we investigate the performance of the quantum gates exposed to different type of noises. We find that some errors may not influence the success of the quantum gates, in contrast, some others may destroy their realization. We show that there is a controlling pattern that can protect quantum gates from certain types of noises and thus can improve the success probability of the gates implementation.
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Abstract We show how to perform measurement-based quantum computing on qudits (high-dimensional quantum systems) using alternative resource states beyond the cluster state. Estimating overheads for gate decomposition, we find that generalizing standard qubit measurement patterns to the qudit cluster state is suboptimal in most dimensions, so that alternative qudit resource states could enable enhanced computational efficiency. In these resources, the entangling interaction is a block-diagonal Clifford operation rather than the usual controlled-phase gate for cluster states. This simple change has remarkable consequences: the applied entangling operation determines an intrinsic single-qudit gate associated with the resource that drives the quantum computation when performing single-qudit measurements on the resource state. We prove a condition for the intrinsic gate allowing for the measurement-based implementation of arbitrary single-qudit unitaries. Furthermore, we demonstrate for prime-power-dimensional qudits that the complexity of the realization depends linearly both on the dimension and the Pauli order of the intrinsic gate. These insights also allow us to optimize the efficiency of the standard qudit cluster state by effectively mimicking more favorable intrinsic-gate structures, thereby reducing the required measurement depth. Finally, we discuss the required two-dimensional geometry of the resource state for universal measurement-based quantum computing. As concrete examples, we present multiple alternative resource states. In certain dimensions, we show the existence of resource states achieving optimal intrinsic gates, enabling more efficient measurement-based quantum information processing than the qudit cluster state and highlighting the potential of qudit stabilizer state resources for future quantum computing architectures.
Measurement-based quantum computation (MBQC) is a framework for quantum information processing in which a computational task is carried out through one-qubit measurements on a highly entangled resource state. Due to the indeterminacy of the outcomes of a quantum measurement, the random outcomes of these operations, if not corrected, yield a variational quantum channel family. Traditionally, this randomness is corrected through classical processing in order to ensure deterministic unitary computations. Recently, variational measurement-based quantum computation (VMBQC) has been introduced to exploit this measurement-induced randomness to gain an advantage in generative modeling. A limitation of this approach is that the corresponding channel model has twice as many parameters compared to the unitary model, scaling as N×DN \times DN×D, where NNN is the number of logical qubits (width) and DDD is the depth of the VMBQC model. This can often make optimization more difficult and may lead to poorly trainable models. In this paper, we present a restricted VMBQC model that extends the unitary setting to a channel-based one using only a single additional trainable parameter. We show, both numerically and algebraically, that this minimal extension is sufficient to generate probability distributions that cannot be learned by the corresponding unitary model.