Session Q49: Quantum Algorithms for Many-Body Systems

Real-Time Dynamics for Computing Hamiltonian Properties

One of the most promising expected applications of near-term quantum computers lies in the study of static and dynamical properties of quantum many-body systems. Many quantum computing algorithms have been proposed with this goal in mind, with a focus on Hamiltonian eigenvalue extraction, a problem central to chemistry, physics, and materials science. However, the majority of established quantum algorithms require a prohibitively large number of resources for near-term hardware. Here we discuss a number of quantum algorithms relying on real-time evolution for energy eigenvalue determination such as quantum Krylov methods and the recently introduced observable-Dynamic Mode Decomposition. Real-time evolution is native to quantum hardware, making these algorithms particularly suited for the near term. We provide strong theoretical and numerical evidence that these methods can converge rapidly even in the presence of noise and demonstrate their efficacies numerically on a range of chemically relevant Hamiltonians.   

Estimating Eigenenergies from Quantum Dynamics: A Unified Noise-Resilient Measurement-Driven Approach

Ground state energy estimation in physics and chemistry is one of the most promising applications of quantum computing. In this paper, we introduce a novel measurement-driven approach that finds eigenenergies by collecting real-time measurements and post-processing them using the machinery of dynamic mode decomposition (DMD). We provide theoretical and numerical evidence that our method converges rapidly even in the presence of noise and show that our method is isomorphic to matrix pencil methods developed independently across various scientific communities. Our DMD-based strategy can systematically mitigate perturbative noise and stands out as a promising hybrid quantum-classical eigensolver.   

Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

We propose a natural application of Quantum Linear Systems Problem (QLSP) solvers such as the HHL algorithm to efficiently prepare highly excited interior eigenstates of physical Hamiltonians in a variational and targeted manner. This is enabled by the efficient computation of the expectation values of inverse Hamiltonians on quantum computers, in situations where Hamiltonian simulation and the representation of eigenstates on quantum computers are efficient. Importantly, the usage of the QLSP solver as a subroutine within our algorithm -- with its inputs and outputs corresponding to physically meaningful objects such as Hamiltonians and eigenstates arising from physical systems -- does not conceal exponentially costly pre/post-processing steps that usually accompanies it in generic linear algebraic applications. We detail implementations of this scheme for both fault-tolerant and near-term quantum computers, analyze their efficiency and implementability, and detail conditions under which the QLSP solvers' exponentially better scaling in problem size render it advantageous over existing classical and quantum approaches. Simulation results for applications in many-body physics and quantum chemistry further demonstrate its effectiveness and scalability over existing approaches.   

Verifiable Solutions to the Schrodinger Equation with Variational Quantum Imaginary-Time Evolution

Variational optimization of parameterized quantum states is a major use of current and future quantum technologies. While recent advancements has resulted in more robust algorithms, several challenges persist in the development of successful variational quantum eigensolvers (VQEs). In particular, parameterized quantum circuits can become stuck in local minima and fail to achieve the ground state, due to barren plateaus in the cost-function landscape. Here we propose a variance-based VQE, which results in verifiable solutions to the Schrodinger equation by direct estimation of the wavefunction variance. We connect previously reported variance-based techniques to the imaginary-time evolution (ITE) formalism. Finally, we show that with little alteration the present algorithm can be used to achieve excited eigenstates. While the measurement cost for the algorithm is a quadratic increase over energy-minimization VQEs, we argue that the verifiability and the simple extension to excited states makes variance-based VQEs a promising technique for finding eigenstates of structured Hamiltonians.   

Subspace-Search for Excited State Quantum Imaginary Time Evolution

Calculating the ground state energies of quantum systems, such as molecules, is an area of expected computational advantage in the noisy intermediate-scale quantum era. Near term variational algorithms, such as the Variational Quantum Eigensolver (VQE), have been developed to determine the ground state energies of these systems. However, excited states have received relatively less attention. We propose a method to determine the excited states of quantum systems through adapting the Subspace-Search method from the Subspace-Search Variational Quantum Eigensolver (SSVQE) to Quantum Imaginary Time Evolution (QITE). As QITE is less susceptible to becoming trapped in local minima, SSQITE may be able to outperform SSVQE for certain systems and anstazes.   

Novel implementation of variational quantum imaginary time evolution for solving general quantum unconstrained binary optimization problems

Quadratic unconstrained binary optimization (QUBO) represents many classical optimization challenges. The Quantum Approximate Optimization Algorithm (QAOA), as well as host of variants, aim to solve these problems on near term quantum computers. We present a new method based on a variational ansatz, coupled with a novel implementation of Variational Quantum Imaginary Time (VarQITE), that solves a variety of QUBO problems with meager cost. Using intuition based on QAOA, we design an ansatz with circuit depth that scales quadratically in qubit size, with parameters that can be optimized straightforwardly via VarQITE, avoiding problem agnostic classical optimizers. Unlike past implementations of VarQITE, updates require only energy evaluations that scale quadratically in qubit number. We present results for weighted MAXCUT problems ranging from 6-20 qubits with different graph types and show our algorithm achieves an approximation ratio of ~1. We find that the number of parameter updates to reach these results scales linearly in qubit number at the sizes we have simulated, and that the results are relatively insensitive to shot noise. Given the modest circuit depth, number of circuit evaluations, and shots required, this work represents a promising avenue for exploring quantum advantage for QUBO problems on near term computers.   

The Quantum Zeno Monte Carlo method for the Hamiltonian eigenstate properties

In this work, we develop the Quantum Zeno Monte Carlo method to find Hamiltonian eigenstate properties such as the ground state energy and the Green's function. Our method is classical-quantum hybrid algorithm and is based on the quantum zeno effect, which is the phenomenon that repeated measurements slow down the speed of transition. Our method finds Hamiltonian eigenstates by implementing the quantum zeno-like procedure using the Monte Carlo method. Unlike popular variational algorithms, our method does not rely on specific ansatz, so it is free of barren plateau problem. Moreover, we show that our method can find Hamiltonian eigenstate properties within a polynomial quantum cost, if the Hamiltonian satisfies moderate conditions. We demonstrate our method in two ways. First, by applying our method for a small system using the quantum processing unit and the noisy simulator, we show that the method is applicable for current NISQ hardware. Second, we verify the polynomial solvability of our method by applying it to the half-filled Hubbard model with various sizes through exact simulator.   

Trotter24: A precision-guaranteed adaptive stepsize Trotterization for Hamiltonian simulations

Choosing an optimal time step is crucial for an efficient Hamiltonian simulation based on Trotterization but difficult due to the complex structure of the Trotter error. In this talk, we present a method measuring the Trotter error without ancillary qubits by combining the second- and fourth-order Trotterizations rather than consulting with mathematical error bounds. Implementing this method, we construct an algorithm, which we name Trotter24, for adaptively using almost the largest stepsize, which keeps quantum circuits shallowest, within an error tolerance preset for our purpose. Trotter24 applies to generic Hamiltonians, including time-dependent ones, and can be generalized to any orders of Trotterization. Benchmarking it in a quantum spin chain, we find the adaptively chosen time step to be about ten times larger than that inferred from known upper bounds of Trotter errors. Trotter24 allows us to keep the quantum circuit thus shallower within the error tolerance in exchange for paying the cost of measurements.   

Quantum simulation of a spin-boson Hamiltonian and its performance.

We aim to expand the range of applications of quantum computers by exploring their potential in investigating effects within cavity quantum electrodynamics (cQED). This includes studying material properties, multi-photon phenomena like superradiance, and systems with strong field-matter interactions. The motivation for developing a quantum computing simulation spans from the fact that experimental studies in cQED are costly, and classical simulations can be challenging.

We are interested in multi-photon and multi-atom setups where physical phenomena such as phase transition in interconnected cavities can be simulated with modest quantum resources. We introduce a boson-to-qubit mapping based on the inverse Holstein-Primakoff transformation which is a promising technique and can be used in other systems driven by a spin-boson, as well as a fermion-boson or a pure boson Hamiltonian.

To conclude the study, we discuss various approaches to time evolution and the effect of the noise when run on real quantum devices.   

Prolonging a discrete time crystal by quantum-classical feedback

Non-equilibrium phases of quantum matter featuring time crystalline eigenstate order have been realized recently on noisy intermediate-scale quantum (NISQ) devices. While ideal quantum time crystals exhibit collective subharmonic oscillations and spatio-temporal long-range order persisting for infinite times, the decoherence time of current NISQ devices sets a natural limit to the survival of these phases, restricting their observation to a shallow quantum circuit. Here we propose a scheme that leverages quantum-classical feedback protocols to enhance a time crystal signal significantly exceeding the decoherence time of the device. As a case of study, we demonstrate the survival of the many-body localized discrete time crystal phase (MBL-DTC) in the one dimensional periodically kicked Ising model, accounting for decoherence of the system with an environment, in a classical simulation. We employ a scheme that uses a periodic quantum-classical feedback protocol based on measurement outcomes obtained in subregions of the system. This approach is suitable for implementation on existing quantum hardware and presents a prospective path to simulate complex quantum many-body dynamics that transcend the low depth limit of current digital quantum computers.   

Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor

In this talk, I show a proof-of-principle demonstration of two medium-term hybrid quantum algorithms for finding ground states, unitary variational quantum phase estimation (UVQPE) and observable dynamical mode decomposition (ODMD). Both algorithms use real-time evolution on a quantum device to generate a small linear algebra problem that is solved classically to find an approximate ground state and corresponding energy. I demonstrate both algorithms on a frustrated 8-spin Heisenberg model, consisting of one "star" of the square kagome lattice, using simulations on the Quantinuum H1 processor and the corresponding noisy emulator. Even in the presence of noise, both algorithms rapidly converge, giving good approximations to the ground state of the model. I furthermore show how, using the spin-symmetry of the Heisenberg model, these algorithms can also be used to compute the magnetization curve.