Session Y37: Quantum Algorithms for Physical Properties Prediction

Variational Quantum Simulations of Multi-Orbital Impurity Models

We perform a systematic study of preparing ground states of correlated multi-orbital impurity models using the variational quantum eigensolver (VQE). We consider both fixed and adaptive wavefunction ans\"atze and analyze the resulting gate depths and performance. We analyze the qubit-adaptive VQE algorithm in the Hartree-Fock orbital basis, as well as the Hamiltonian variational ansatz (HVA) and a variant of adaptive VQE in the atomic orbital basis. 

An operator pool composed of pairwise commutators of the Hamiltonian terms is developed to allow a fair comparison between the adaptive and the fixed HVA ansatz. Using state vector simulations, we show that the most compact ans\"atze is obtained in the atomic orbital representation with symmetry-based Pauli tapering within parity encoding. Finally, we perform adaptive VQE calculations including sampling noise and demonstrate that using a doubly decomposed form of the impurity Hamiltonian, symmetry-based qubit reduction, and stochastic optimizers dramatically reduces the number of shots required for the desired accuracy.   

Adaptive variational algorithms for quantum Gibbs state preparation

The preparation of Gibbs states is an important task in quantum computation with applications in quantum simulation, quantum optimization, and quantum machine learning. However, many algorithms for preparing Gibbs states rely on quantum subroutines which are difficult to implement on near-term hardware. Here, we address this by (i) introducing an objective function that, unlike the free energy, is easily measured and (ii) using dynamically generated, problem-tailored ansatze. This allows for arbitrarily accurate Gibbs state preparation using low-depth circuits. To verify the effectiveness of our approach, we numerically demonstrate that our algorithm can prepare high-fidelity Gibbs states across a broad range of temperatures and for a variety of Hamiltonians.   

Can quantum natural gradient improve variational quantum algorithms?

Natural gradients have been used to improve convergence of the classical variational Monte Carlo methods to simulate molecular Hamiltonians. Recently a quantum genralization of the natural gradient approach was proposed based on the quantum Fisher information. We employ quantum circuits to compute the components of the Fisher information matrix and study the contribution of diagonal and off-diagnoal terms on the convergence of variation quantum eigensolver by considering two different classes of quantum chemistry Hamiltonians.    

Making predictions in a quantum world

Many scientific advancements in physics and chemistry depend on our ability to learn and make predictions in a quantum-mechanical world. In this talk, I will review recent advances in understanding what we could learn from quantum experiments and how efficient could we be. I will present results showing that sometimes we could make reliable predictions using much fewer experiments than one may expect and provide relevant problems that are actually impossible to learn. Finally, I will give rigorous theorems and proof-of-principle experiments showing how the development of quantum technology, including a combination of quantum sensors, quantum memory, and quantum computers, could significantly enhance our ability to learn and predict.

The talk will cover results from collaborations with John Preskill, Richard Kueng, Jordan Cotler, Sitan Chen, Jerry Li, Jarrod McClean, Michael Broughton, Steve Flammia, and others.   

Contracted Quantum Eigensolvers for Quantum Simulation

Simulating many-body quantum systems poses a significant challenge and opportunity for near-term quantum computing. Here, we highlight a class of recently developed quantum contracted eigensolvers. We focus on an approach which aims to solve the anti-Hermitian part of the contracted Schrodinger equation (ACSE) on a quantum computer. Our method exhibits exponential cost reductions on a quantum computer, and leads to iterations with only polynomial scaling tomography in the construction of the reduced density matrices. We also highlight applications of a qubit-particle parametrization of a fermionic wavefunction. This allows for a reduction in the multi-qubit gate cost in our solution of the ACSE, while still maintaining a high level of accuracy. Finally, we explore applications of classical optimization techniques to the qACSE approach, and are able to demonstrate rapid convergence towards a solution of the ACSE across a variety of systems.   

A quantum-classical eigensolver using multiscale entanglement renormalization

Strongly-correlated quantum many-body systems are very hard to study and simulate classically. Based on the multi-scale entanglement renormalization ansatz (MERA) and gradient-based optimization, we propose a variational quantum eigensolver (VQE) for the simulation of strongly correlated quantum matter. Compared to corresponding classical algorithms with large tensor contraction costs, this MERA quantum eigensolver has significantly lower computation costs. This algorithm is capable of being implemented on NISQ devices while still describing very large systems because of its narrow causal cone. This feature is particularly attractive for ion-trap devices with ion-shuttling capabilities. As the size of the simulated system grows, the total number of qubits required grows logarithmically, but the number of qubits needed in the interaction region is system-size independent. Translation invariance of the simulated systems can be used to make computation costs square-logarithmic in the system size and describe the thermodynamic limit. We demonstrate the approach numerically for a MERA with Trotterized disentanglers and isometries. With a few Trotter steps, one recovers the accuracy of the full MERA.   

Implementing Translational Quantum Subspace Expansion with Fewer Qubits

Translational Quantum Subspace Expansion (TransQSE) is a hybrid quantum-classical algorithm for finding ground states of periodic systems by variationally exploring correlations within a finite sector of space, then mixing correlations across sectors via QSE. The original formulation requires 2N+1 qubits, where N is the number of orbitals within one sector. Applying our own ansatz and a qubit mapping developed for single-body band structure calculations, we implement a modified version of TransQSE requiring fewer qubits. We compare the performance of our implementation with the original formulation, subject to both ideal and noisy environments.   

Quantum pixel representations and compression for N-dimensional images

We present a novel and uniform framework for quantum pixel representations that overarches many of the most popular representations proposed in the recent literature, such as (I)FRQI, (I)NEQR, MCRQI, and (I)NCQI. The proposed QPIXL framework results in more efficient circuit implementations and significantly reduces the gate complexity for all considered quantum pixel representations. Our method only requires a linear number of gates in terms of the number of pixels and does not use ancilla qubits. Furthermore, the circuits only consist of Ry gates and CNOT gates making them practical in the NISQ era. Additionally, we propose a circuit and image compression algorithm that is shown to be highly effective, being able to reduce the necessary gates to prepare an FRQI state for example scientific images by up to 90% without sacrificing image quality. Our algorithms are made publicly available as part of QPIXL++, a Quantum Image Pixel Library.   

Quantum Simulation of Dihedral Gauge Theories

In this talk, I describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group DN – the dihedral group – serves as an approximation to U(1) × Z2 lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the nonabelian Fourier transform over DN, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in n = log N. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of D4. The fidelity of all D4 gates was found to exceed 80%.   

A Quantum Algorithm to Simulate Open Quantum Systems

Given the advent of quantum algorithms for a wide array of problems in linear algebra and machine learning, it is important to develop general methods for the simulation of arbitrary (i.e. non-unitary) operators on quantum hardware.  In this talk, we present a novel quantum algorithm based on the quantum singular value transformation (QSVT) to apply an arbitrary operator K to some input state and subsequently estimate the expectation value of some observable.  Our construction then immediately yields a route to estimating observables of states undergoing open quantum dynamics, whose effect is captured by a set of non-unitary Kraus operators.  Our algorithm succeeds deterministically given the Sz-Nagy dilation, and we provide details on the algorithm's query and gate complexity, numerical verification, and comparisons with prior methods.   

Variational Quantum-Neural Hybrid Eigensolver

The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-size quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some non-trivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this work, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum ansatz can be further enhanced by classical post-processing with neural networks. We show that VQNHE consistently and significantly outperforms VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that for arbitrary post-processing neural functions, VQNHE only incurs an polynomial overhead of processing time and represents the first scalable method to exponentially accelerate VQE with non-unitary post-processing that can be efficiently implemented in the NISQ era.   

Dimensional Reduction in Quantum-Enhanced Stochastic Modelling

In data analytics, the curse of dimensionality is a well-acquainted adversary. As we seek to make predictions from time-series data drawn from processes of ever-growing complexity, modelling the possible future effects from all possible past observations becomes quickly intractable. Even when the time-series data is binary, the cost of accounting for temporal correlations in the last n time-steps grows as 2ⁿ – making the exact simulation of highly non-Markovian processes computationally infeasible. In this talk, we describe how quantum models – machines the store relevant past information in quantum memory - has the potential to significantly outperform their classical counterparts. Notably:

1. Given models of a fixed memory dimension, quantum models can achieve superior accuracy than their classical counterpart

2.  There exist families of progressively more non-Markovian processes that require increasing classical memory dimensionality to model, and yet can be modelled by a quantum machine of bounded dimension.

We illustrate such quantum models discovered directly from time-series data, and how they can display provable accuracy advantage within today’s noisy quantum processors. We discuss how such models can also generate future predictions in a quantum superposition, providing a key sub-routine for various quantum algorithms that enable the enhanced analysis of stochastic processes (e.g., quantum amplitude estimation, risk analysis, importance sampling).   

Efficient Quantum Computation of Floquet Hamiltonians

The Floquet formalism describes the control over quantum systems using external periodic fields. With recent advances in ultrafast spectroscopy of solid-state systems, Floquet engineering, that is, a targeted design of quantum systems driven by laser pulse, has led to an increasing interest in computational methods that can simulate light-matter interactions. Although the perturbative regime, in which the fundamental driving frequency is much larger than the energy bandwidth of the quantum system, shows interesting phenomena, it is the non-perturbative regime that presents the most exciting opportunity to study the interplay with strong correlations and which remains largely unexplored. Here we describe hybrid quantum algorithms that make use of quantum computers to tackle this problem. The required quantum resources are within reach for current day NISQ devices and allow the efficient computation of Floquet Hamiltonians. We demonstrate applications of these algorithms and discuss their performance for small scale driven quantum systems.