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arXiv:2210.06923 · Physical Review A

Imaginary-time evolution with quantum nondemolition measurements: Multiqubit interactions via measurement nonlinearities

Manikandan Kondappan, Manish Chaudhary, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, Tim Byrnes

Published Apr 20, 2023• 14 citations
arXivHTMLPDF

Abstract

We show that quantum nondemolition (QND) measurements can be used to realize measurement-based imaginary-time evolution. In our proposed scheme, repeated weak QND measurements are used to estimate the energy of a given Hamiltonian. Based on this estimated energy, adaptive unitary operations are applied such that only the targeted energy eigenstates are fixed points of the evolution. In this way, the system is deterministically driven towards the desired state. The nonlinear nature of the QND measurement, which allows for producing interactions between systems, is explicitly derived in terms of measurement operators. We show that for suitable interaction times, single-qubit QND Hamiltonians can be converted to effective multiqubit imaginary time operations. We illustrate our techniques with the example of preparing a four-qubit cluster state, which is prepared using only collective single-qubit QND measurements and single-qubit adaptive operations.

DOI
10.1103/physreva.107.042616

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We show that quantum nondemolition (QND) measurements can be used to realize measurement-based imaginary-time evolution. In our proposed scheme, repeated weak QND measurements are used to estimate the energy of a given Hamiltonian. Based on this estimated energy, adaptive unitary operations are applied such that only the targeted energy eigenstates are fixed points of the evolution. In this way, the system is deterministically driven towards the desired state. The nonlinear nature of the QND measurement, which allows for producing interactions between systems, is explicitly derived in terms of measurement operators. We show that for suitable interaction times, single-qubit QND Hamiltonians can be converted to effective multiqubit imaginary time operations. We illustrate our techniques with the example of preparing a four-qubit cluster state, which is prepared using only collective single-qubit QND measurements and single-qubit adaptive operations.
We introduce a method to perform imaginary time evolution in a controllable quantum system using measurements and conditional unitary operations. By performing a sequence of weak measurements based on the desired Hamiltonian constructed by a Suzuki-Trotter decomposition, an evolution approximating imaginary time evolution can be realized. The randomness due to measurement is corrected using conditional unitary operations, making the evolution deterministic. Both the measurements required for the algorithm and the conditional unitary operations can be constructed efficiently. We show that the algorithm converges only below a specified energy threshold and the complexity is estimated for some specific problem instances.
We discuss monitoring the time evolution of an analog quantum simulator via a quantum nondemolition (QND) coupling to an auxiliary "clock" qubit.The QND variable of interest is the "energy" of the quantum many-body system, represented by the Hamiltonian of the quantum simulator.We describe a physical implementation of the underlying QND Hamiltonian for Rydberg atoms trapped in tweezer arrays using laser-dressing schemes for a broad class of spin models.As an application, we discuss a quantum protocol for measuring the spectral form factor of quantum many-body systems, where the aim is to identify signatures of ergodic versus nonergodic dynamics, which we illustrate for disordered one-dimensional Heisenberg and Floquet spin models on Rydberg platforms.Our results also provide the physical ingredients for running quantum phase estimation protocols for measurement of energies and preparation of energy eigenstates for a specified spectral resolution on an analog quantum simulator.
Abstract Imaginary-time evolution plays an important role in algorithms for computing ground-state and thermal equilibrium properties of quantum systems, but can be challenging to simulate on classical computers. Many quantum algorithms for imaginary-time evolution have resource requirements that are prohibitive for current quantum devices and face performance issues due to noise. Here, we propose a new algorithm for computing imaginary-time evolved expectation values on quantum computers, inspired by probabilistic error cancellation, an error-mitigation technique. Our algorithm works by decomposing a Trotterization of imaginary-time evolution into a probabilistic linear combination of operations, each of which is then implemented on a quantum computer. The measurement data is then classically post-processed to obtain the expectation value of the imaginary-time evolved state. Our algorithm requires no ancillary qubits and can be made noise-resilient without additional error-mitigation. It is well-suited for estimating thermal expectation values by making use of the notion of a thermal pure quantum state. We demonstrate our algorithm by performing numerical simulations of thermal pure quantum state preparation for the 1D Heisenberg Hamiltonian on 8 qubits, and by using an IBM quantum computer to estimate the energy of the same Hamiltonian on 2 qubits. We observe promising results compared to the exact values, illustrating the potential of our algorithm for probing the physics of quantum many-body systems on current hardware.
Algorithms based on non-unitary evolution have attracted much interest for ground state preparation on quantum computers. One recently proposed method makes use of ancilla qubits and controlled unitary operators to implement weak measurements related to imaginary-time evolution. In this work we specialize and extend this approach to the setting of combinatorial optimization. We first generalize the algorithm from exact to approximate optimization, taking advantage of several properties unique to classical problems. In particular we show how to select parameters such that the success probability of each measurement step is bounded away from 1/21/21/2. We then show how to adapt our paradigm to the setting of constrained optimization for a number of important classes of hard problem constraints. For this we compare and contrast both penalty-based and feasibility-preserving approaches, elucidating the significant advantages of the latter approach. Our approach is general and may be applied to easy-to-prepare initial states as a standalone algorithm, or deployed as a quantum postprocessing stage to improve performance of a given parameterized quantum circuit. We then propose a more sophisticated variant of our algorithm that adaptively applies a mixing operator or not, based on the measurement outcomes seen so far, as to speeds up the algorithm and helps the system evolution avoid slowing down or getting stuck suboptimally. In particular, we show that mixing operators from the quantum alternating operator ansatz can be imported directly, both for the necessary eigenstate scrambling operator and for initial state preparation, and discuss quantum resource tradeoffs.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ``deterministic randomness'' in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity.