Session A50: Progress in Quantum Simulation

The optimal depth of variational quantum algorithms is hard to compute, even approximately

Abstract: Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ”ansatz” used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ε > 0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N(1−ε), for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even ”simpler” QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.

This talk assumes no background in Computer Science and/or Complexity Theory.   

Tradeoffs in hybrid quantum-classical algorithms for designing quantum optimal controls

Quantum control holds promise for impacting fundamental research in physics, chemistry, and materials science. A persistent challenge in this field stems from the prohibitive cost of simulations, which are needed to inform quantum control advances in the laboratory. In this talk, we explore the costs and tradeoffs of performing quantum optimal control simulations based on the GRAPE and CRAB algorithms, when these algorithms are formulated as hybrid quantum-classical algorithms. In this setting, a quantum computer is responsible for performing the quantum dynamics simulations needed to evaluate the control objective functional, and potentially gradients thereof, via time-dependent Hamiltonian simulation. We will outline costs and tradeoffs associated with time-dependent Hamiltonian simulation using Trotter and post-Trotter methods, and contextualize this discussion with recent results obtained empirically through numerical simulation.

SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.   

Sequential quantum simulation with circuit QED devices

Quantum simulation of many-body systems in materials science and chemistry are promising application areas for quantum computers. However, the limited scale and coherence of near-term quantum processors pose a significant obstacle to realizing this potential. Here, we outline how circuit quantum electrodynamics (cQED) devices, each consisting of a transmon qubit coupled to a long-lived cavity mode, can be used to simulate the ground state of a highly-entangled quantum many-body spin chain. We exploit recently developed methods for implementing quantum operations to sequentially build up a matrix product state (MPS) representation of a many-body state. This approach re-uses the transmon qubit to read out the state of each spin in the chain and exploits the large state space of the cavity as a quantum memory encoding inter-site correlations and entanglement.

We show that analog (pulse-level) control schemes can accurately prepare a known MPS representation of a quantum critical spin chain in significantly less time than digital (gate-based) methods, thereby reducing the exposure to decoherence. We then explore this analog-control approach for the variational preparation of an unknown ground state. Practically, the large state space of the cavity can be used to replace multiple qubits in a qubit-only architecture, and could therefore simplify the design of quantum processors for simulating a wide range of materials, from the Ising model to fractional quantum Hall systems.   

Towards Quantum Simulation of Confining Gauge Theories at Finite Temperature and Density

Numerically simulating strongly coupled gauge theories at finite density is a

longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. Recently it has been suggested that digital and analog quantum hardware could provide pathways to efficiently simulate such systems. In this work, we adapt numerical tools based on imaginary time evolution, originally developed to simulate strongly correlated condensed matter systems, to study this problem. We do so in the simplest model of a confining gauge theory, namely $mathbb{Z}_2$ gauge theory coupled to spinless fermionic matter in 1+1 dimensions, which can be directly mapped to a local, interacting quantum spin chain. We discuss classical numerical results (from both sparse matrix and matrix product state calculations) on the finite-temperature and -density equation of state, as well as the chiral and confinement phase diagrams. We comment on how the relevant observables can be obtained with a recently proposed quantum algorithm, adaptive variational quantum minimally entangled typical thermal states (AVQMETTS), and when this approach is expected to outperform classical methods. Our work sets the stage for new approaches to study strongly coupled gauge theories on quantum hardware that can be generalized to higher dimensions where classical numerics become severely limited.   

Calculation of nucleon-pair transition intensities on a quantum computer

We simulated the calculation of nucleon-pair transition intensities between systems with Ω+1 and Ω+2 nucleon pairs around the critical point of the transition between vibrational and rotational phases on an ideal quantum computer. To this end we mapped the bosonic two-level Hamiltonian, describing the motion of nucleon pairs in two j-shells with identical pair degeneracy Ω, into the Hilbert space of the quantum device. The unitary matrices transforming the initial state of the quantum computer into the ground states of the systems involved in the nucleon-pair transition were obtained using the Variational Quantum Eigensolver algorithm with simple quantum circuits consisting of two layers of RY rotations, and a fully connecting layer of CZ gates in between them. Combining these matrices with the transition operator defined in the two-level model yielded a chain of Pauli strings that was measured on the quantum computer's initial state to calculate the two-nucleon transition intensity. Initial results are in good agreement with classical ones up to a pair degeneracy Ω=15. Beyond this degeneracy, the transition intensities agree with classical results qualitatively.   

Idealistic dynamical simultions of quantum multi-dimensional quantum-manybody systems

WIthin recent years there has been a huge push by computational condensed matter researchers to implement the use of quantum computers within their studies. Computational condensed matter researchers spend a lot of time looking at quantum many-body systems and their properties by building simulations, but due to the current hardware limitations with the best of our quantum computers and the faster numerical handling of data, a switch to quantum computers seems like a natural choice. With the rise and rapid growth in development of quantum hardware, it leaves us to question the limit of the hardware and if is can be used to study complex and less established condensed matter models.In 2019 a small team at IBM aimed to answer this question by studying well known 1 dimensional spin spin models, and found that it if we can find an appropriate mapping of the model into the quantum computer, then we can use a quantum computer to build simulations and study quantum many body systems. This was a big development for the field of computational condensed matter physics, but this does not encompass a big group of researchers studying multi-dimensional quantum many body systems. With this project I aim to test the limits of the use of quantum computers within condensed matter research by using them to study well known and established spin systems in multiple dimensions, including spin direction and lattice dimensionality. I then will compare the results from the quantum computer to classical simulations of the same model.   

Noise-Resilient Trial-State Optimization Algorithm for Quantum Gap Estimation

Quantum simulation is believed one of the most promising avenues to reveal quantum advantage. Quantum phase estimation is a paradigm of unbiased quantum simulation but, in the original form, with high demand for deep circuits to tolerate errors. Hybrid quantum/classical algorithms such as variational methods were proposed as alternative approaches using shallow circuits on contemporary noisy quantum hardware while not offering a significant prospect for quantum advantage in scale up. In this work, we revisit the hybrid quantum gap estimation (QGE) algorithm [1] that combines Trotterized time evolution on a quantum processor with the classical fast Fourier transform of the damped time propagator to estimate the energy gaps of a many-body model to within a certain tolerance. We improve the accuracy of QGE by implementing trial-state unitaries with optimization parameters and using a classical optimizer to maximize the peak-to-background ratio in the many-body spectral function [2]. To verify performance, we demonstrate our algorithm for an example many-body model on the IBMQ simulator using device noise models. We prove that our algorithm is resilient against common mid-circuit Markovian noise channels, thus opening the door to a significant performance boost in scale up.   

Quantum imaginary time evolution on 1D infinite-size lattice

I will show how to implement the quantum imaginary time evolution (QITE) and QLanczos algorithms on a 1D infinite-size lattice. We transform the uniform matrix product state (uMPS) to the quantum circuit and use the concept of time-dependent variational principle (TDVP) to implement QITE. The transverse-filed Ising model simulation results both on a simulator and on IBM-Q real devices will be shown. We will compare the simulation results from the classical tensor network TDVP algorithm and those from quantum circuit.   

Quantum digital simulation of the nuclear de-excitation of tritium

The description of nuclear processes requires knowledge of the quantum state of the nucleus, which is often difficult to address analytically. Quantum computing algorithms can prepare the qubits in such a way that encodes the state of the nuclei and also by simulating the action of observables on such states. We show that quantum computers may embed a full simulation. Here we show the pipeline of processes that allow to simulate a nuclear transition from state preparation to photon emission. It is already known how to prepare quantum states of a deuterium nucleus by using a quantum computer, as well as how to evaluate the probability of the transition between such states. If the quantum states are not known a priori, the most common algorithms that find them are the variational quantum eigensolver for the ground state and its variations for the excited states, respectively. We combine the methods so to determine the ground state and first excited state by using hybrid quantum-classical computing. We estimate a relative error of ∼ 2% for the ground state energy eigenvalue and ∼ 10% for the first excited state, respectively. The transition probability between the two levels reaches the maximum for dipole polarization angle at around 2.8 rad. This work is a first step towards a quantum digital simulation pipeline in nuclear physics. Our work can be extended to other quantum systems, such as quantum chemistry, atomic physics and photonics.   

Realizing a parametrically coupled lattice of superconducting qubits for quantum simulation with a synthetic magnetic field

Advances in quantum computing hardware have made possible the burgeoning field of quantum simulation of condensed matter systems. Arrays of superconducting qubits, in particular, offer native emulation of lattice physics together with a high degree of tunability. Emulation of magnetic fields, however, is a challenge in this platform, yet magnetic fields are a central ingredient in many models of interest. Here, we introduce a scheme to emulate the dynamics of charge carriers through a magnetic field by parametrically driving hopping of microwave photons between transmon qubits. Implementing our scheme in a 4-by-4 array of qubits, we emulate arbitrary magnetic fields with uniform and disordered field profiles. We verify key signatures of the presence of a synthetic magnetic field, and we discuss challenges to implementing the scheme on real-world hardware.   

Photon-Mediated Interactions in Quasi-1D Lattices of Coplanar Waveguide Resonators

Circuit quantum electrodynamics (circuit QED) is quickly becoming one of the main platforms for quantum simulation and computation. One of its notable advantages is its ability to facilitate the study of new regimes of light-matter interactions. This is achieved due to the native strong coupling between superconducting qubits and microwave resonators, and the ability to lithographically define a large variety of resonant microwave structures, for example photonic crystals. Such geometries allow the implementation of novel forms of photon-mediated qubit-qubit interaction, and studies of many-body physics. In this talk, I will introduce how coplanar waveguide (CPW) lattices can be used to create engineered photon-mediated interactions. I will present preliminary data characterizing photon-mediated interactions between tunable transmon qubits in a quasi-1D lattice of CPW resonators with unconventional bands, such as gapped and ungapped flat bands.   

Compressing quantum time evolution circuits of impurity models

Simulation of impurity models, i.e. many body systems with sparse interactions, allows us to study the physics of various materials. While some systems such as the Kondo model are directly represented as an impurity model, other more general systems (e.g. the Hubbard model) can make use of impurity models in embedding methods as DMFT and cluster DMFT. When the number of impurities is small, these models can be solved using conventional computational methods. However, the cost of these methods increases exponentially with the number of impurities, and the need for quantum computation emerges. By using the Trotter-Suzuki formula, quantum circuits for time evolution of these models can be generated with circuit depth scaling as O(n*t) where n is the system size and t is the number of time steps. This results in deep circuits, which are not amenable to near-term and early fault tolerant hardware. To reduce the circuit depth, here we use a recently introduced algebraic compression method, as well as its extension to long range fermion hopping. The method is based on local circuit operations of fusion, commutation and turnover, that are satisfied by the free fermion terms. By using these operations, we show that the time evolution circuits can be compressed down to O(d*t) depth where d is the number of impurities, and t is the number of time steps. We demonstrate our method by simulating the RKKY model to obtain an effective Hamiltonian for two spins in a metal.   

Hadron dynamics in one dimensional quantum electrodynamics using digital quantum computers

The dynamics of composite particles, ``hadrons", is simulated in one-dimensional lattice quantum electrodynamics using digital quantum computers. This simulation involves the local creation of hadrons on top of the vacuum, which are evolved forward in time. The preparation of the initial state on an array of qubits is done with the SC-ADAPT-VQE algorithm recently developed by the authors. This work uses charge screening and the Lieb-Robinson bound to develop efficient and scalable techniques for time evolution. End-to-end simulations of hadron dynamics are performed on IBM’s superconducting quantum computer and compared to classical tensor network simulations.