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Decoherence Limits the Cost to Simulate an Anharmonic Oscillator, arXiv:2307.00748

In this paper, Tzula B. Propp, Sayonee Ray, John B. DeBrota, Ivan Deutsch, and myself study how decoherence in the open system Kerr model makes classical simulation of the dynamics using finite-difference numerical integration less costly. We also show that the expectation values of observables only significantly deviates from those calculated using the truncated Wigner approximation (TWA), corresponding to the semi-classical dynamics of the system, at specific time intervals corresponding to when the symmetries of the observable and the corresponding kitten state match. Otherwise, we find the TWA gives an excellent approximation, which gets even better with time in the presence of decoherence.

Diabatic Quantum Annealing for the Frustrated Ring Model, arXiv:2212.02624

In this paper, Jeremy Côté, Frédéric Sauvage, Martin Larocca, Matías Jonsson, Lukas Cincio, and myself show that by optimizing the annealing schedule of quantum annealing and allowing for diabatic transitions, we are able to solve a simple Ising problem exhibiting an exponentially closing (perturbative) gap, known as the frustrated ring model. One nice aspect of this 1-dimensional system that lends itself nicely for optimizing the annealing schedule is that we are able to simulate relatively large system sizes by only evolving the expectation value of the problem Hamiltonian. The optimized annealing schedules we found can be found in a Zenodo repository here.

Quantum-Inspired Tempering for Ground State Approximation using Artificial Neural Networks, arXiv:2210.11405, SciPost Phys. 14, 121 (2023)

In this paper, Conor Smith, Quinn Campbell, Andrew Baczewski, and myself propose a new method to augment the standard training of artificial neural networks (ANNs) ansatze to approximate ground state of many-body quantum Hamiltonians. Our method is inspired by quantum parallel tempering; different replicas are evolved independently with different Hamiltonians using the standard Stochastic Reconfiguration method, but the replicas are then allowed to swap configurations with a certain probability. We show that this new method can improve on the standard approach when the training is likely to get stuck in local minima.

On the Emerging Potential of Quantum Annealing Hardware, arXiv:2210.04291

In this paper, Byron Tasseff, Zachary Morrell, Marc Vuffray, Andrey Lokhov, Sidhant Misra, Carleton Coffrin, and myself study the performance of the D-Wave quantum annealing hardware against several implementations of classical optimization algorithms on a particular class of problems native to the D-Wave connectivity graph, called the Pegasus graph. At the largest problem size we study, corresponding to a Pegasus lattice size of 16, the D-Wave hardware is able to reach a particular approximation of the ground state faster than our other implementations. While THIS IS NOT A DEMONSTRATION OF A QUANTUM ADVANTAGE, it illustrates the potential value of specialized analog hardware at providing wall clock time benefits. Ultimately though, the D-Wave hardware is unable to reach below a certain approximation of the ground state, which we attribute to the effect of noise on performance.

My hope is that this work will motivate the community to make openly available high quality solvers to challenge results from special purpose hardware. I provide a link here to download a zip file that includes all the instances as json files that were tested for the data presented in the manuscript. The Ising Hamiltonian is assumed to be of the form:

\[ H = \sum_{i=1}^n h_i \sigma_i^z + \sum_{i=1}^n \sum_{j=i+1}^n J_{ij} \sigma_i^z \sigma_j^z \]

The python script can be used to see how to load the json files and extract the local fields and couplings. It can be executed for example with:

python -f ./datasets/Pegasus-Lattice_Size-2/Pegasus-Lattice_Size-2_00001.json

where it will output the energy of a random spin configuration for instance Pegasus-Lattice_Size-2_00001. The largest system size corresponds to a size 16 Pegassus graph, and the csv file results.csv includes the list of lowest energies found for the 50 instances using PT-ICM.

Master Equation Emulation and Coherence Preservation with Classical Control of a Superconducting Qubit, arXiv:2210.01388, Phys. Rev. A 106, 062620 (2022)

In this paper, Evangelos Vlachos, Haimeng Zhang, Vivek Maurya, Jeffrey Marshall, Eli M. Levenson-Falk, and myself investigate how to emulate generalized Markovian master equations using stochastic classical driving in the Hamiltonian. We realize this experimentally in superconducting qubits, and we find very good agreement between numerical simulations using the Stochastic Schrödiner Equation and the experimental results. We also demonstrate how introducing a generalized Markovian master equation in addition to a Markovian master equation (along orthogonal axis) can be used to improve the coherence time of the qubit, which is an experimental realization of the work of Jeffrey Marshall, Lorenzo Campos Venuti, and Paolo Zanardi.

Signatures of Open and Noisy Quantum Systems in Single-Qubit Quantum Annealing, arXiv:2208.09068, Phys. Rev. Applied 19, 034053 (2023)

In this paper, Zachary Morrell, Marc Vuffray, Andrey Lokhov, Andreas Bartschi, Carleton Coffrin, and myself investigate different noise models for single qubits on D-Wave's quantum annealing hardware. We show that using the standard annealing protocol we cannot distinguish between an open system model and a closed system model with longitudinal magnetic field noise when one only has access to the final time computational basis statistics. However, if we use D-Wave's h-gain functionality, which allows us to turn off the longitudinal field at intermediate points in the anneal, we can clearly show that only the open system model with longitudinal magnetic field noise can reproduce the statistics of the device.

High-quality Thermal Gibbs Sampling with Quantum Annealing Hardware, arXiv:2109.01690, Phys. Rev. Applied 17, 044046 (2022)

In this paper, Jon Nelson, Marc Vuffray, Andrey Y. Lokhov, Carleton Coffrin, and myself investigated the performance of a D-Wave 2000Q quantum annealing processor as a classical Gibbs sampler. On hand-picked \(J_{ij} = \pm 1\) instances of up to 16 qubits, we were able to identify a ‘sweet spot’ in terms of the programmed Ising strength where the quality of samples was best, and we corroborated this sweet spot with the point where spurious \(ZZ\) couplings are absent. Here, spurious \(ZZ\) couplings correspond to interactions that appear in the effective diagonal Hamiltonian that best describes the observed data but are not interactions that we have programmed. For example, these spurious couplings would capture any residual effect of the transverse field that may be present in the output statistics. While it is not necessarily surprising that the sampling would be best when these spurious coupling are minimized, what is interesting is why a sweet spot such as this would arise at all. We propose a model that includes a residual transverse field and fluctuating local fields that appears to explain the sweet spot very well; in this model the effects of the residual transverse field and the fluctuating local fields cancel each other out in the sweet spot regime. This gives a nice demonstration of the importance of fluctuating noise in understanding the statistical output of the D-Wave 2000Q processor.

3-Regular 3-XORSAT Planted Solutions Benchmark of Classical and Quantum Heuristic Optimizers, arXiv:2103.08464, Quantum Sci. Technol. 7 025008 (2022)

In this paper, Matthew Kowalsky (USC), Itay Hen (USC), Daniel A. Lidar (USC), and myself use the class of 3-regular 3-XORSAT to benchmark several different approaches to solving for the ground state of Ising Hamiltonians. Our suite of solvers includes Parallel Tempering, Fujitsu's Digital Annealer, Toshiba's Simulated Bifurcation Machine, and the Virtual MemComputing Machine, and the D-Wave Advantage quantum annealer. All algorithms, despite important implementation differences, exhibit an exponential growth in the computational cost to solve this class of problems, with Fujitsu's Digital Annealer exhibiting the lowest wall clock time among this suite of solvers.

The instances used in this study can be downloaded here. The format of the instances is as follows:

  • The instance filenames are of the form: 3r3x_2body_nX_sY.txt, where X is the size of the instance and Y is the label of the instance. X takes values in [16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 384, 400, 416, 432, 448, 512, 1024, 2048, 4096] whereas Y takes values from 1 to 100.

  • Each instance file is made of 3 columns, and each line corresponds to an interaction term in the Ising Hamiltonian. The first 2 columns are the spin indices (the indexing starts at 1), and the 3rd column is the value of the interaction. If the two column values are equal, then this corresponds to a local field term \(h_i\) associated with that spin index. If the two column values are not equal, then this corresponds to an Ising coupling \(J_{ij}\) between those two spins. We assume the Ising Hamiltonian has the form:

\[ H = \sum_{i=1}^n h_i \sigma_i^z + \sum_{i=1}^n \sum_{j=i+1}^n J_{ij} \sigma_i^z \sigma_j^z \]

  • The ground state energy of each instance should be equal to \(-2 n\), where \(n\) is the number of spins in the instance (equal to X above).

Customized quantum annealing schedules, arXiv:2103.06461, Phys. Rev. Applied 17, 044005 (2022)

In this paper, Mostafa Khezri (USC), Xi Dai (Waterloo), Rui Yang (Waterloo), Adrian Lupascu (Waterloo), Daniel A. Lidar (USC), and myself study the important problem of how to reverse-engineer an annealing schedule into circuit flux biases for capacitively shunted flux qubits. This problem is non-trivial since the low energy subspace of the flux qubit changes during the annealing protocol, and at every step one has to identify what the effective description of the system is in terms of a transverse field Ising model. Being able to customize the annealing schedule can be useful for example if we want to slow the annealing protocol around the minimum gap, assuming we know where it is located.

Diagonal Catalysts in Quantum Adiabatic Optimization, arXiv:2009.05726, Phys. Rev. A 103, 022608 (2021)

In this paper, Matthew Kowalsky and I show how we can use a diagonal catalyst to reproduce the exponential improvement of adiabatic reverse annealing for the ferromagnetic \(\mbox{p}\)-spin model. Our work highlights the role of the catalyst in generating this improvement: it serves to bias the energy landscape towards the target spin configuration. The success of the protocol also makes clear how it can fail: biasing the energy landscape towards a state only helps in finding the ground state if the Hamming distance from the ground state and the energy of the biased state are correlated. We present examples where biasing towards low energy states that are nonetheless very far in Hamming distance from the ground state can severely worsen the efficiency of the algorithm compared to the standard protocol.

Comparing relaxation mechanisms in quantum and classical transverse-field annealing, arXiv:2009.04934, Phys. Rev. Applied 15, 014029 (2021)

In this paper, Jeffrey Marshall and I study to what extent the ground state repopulation associated with pausing during the anneal can be attributed to the quantum nature of the thermal relaxation process. We show that we can reproduce many of the features of the output of the D-Wave 2000Q quantum annealing processor and the adiabatic master equation with spin-vector Monte Carlo, but we also highlight important qualitative differences between the three.

De-Signing Hamiltonians for Quantum Adiabatic Optimization, arXiv:2004.07681, Quantum, 4, 334 (2020)

In this paper, Elizabeth Crosson, Itay Hen, Peter Young, and I study to what extent a non-stoquastic intermediate Hamiltonian can help improve the performance of the quantum adiabatic optimization algorithm relative to two equivalent stoquastic intermediate Hamiltonian. We compare the spectral gaps between the non-stoquastic and stoquasticized Hamiltonians, and we find that non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization.

Testing a quantum annealer as a quantum thermal sampler, arXiv:2003.00361, ACM Transactions on Quantum Computing, 2(2) (2021).

In this paper, Zoe Izquierd, Itay Hen, and I study to what extent a D-Wave quantum annealing processor can be used as a quantum thermal sampler. This notion of using a quantum annealer as a thermal sampler relies on two key assumptions: (1) that the device if paused at a particular point in its annealing schedule, thermalizes to the quantum thermal state associated with the instantaneous Hamiltonian at that point (the target Hamiltonian), and (2) that the measurement outcomes generated by the device correspond to measurements performed on the quantum thermal state. In this work, we show that the measurement protocol used, which is to 'quench’ from the target Hamiltonian to the measurement Hamiltonian, violates the second assumption, at least for the 1-dimensional transverse field Ising model. We basically find that the 'quench’ is too slow, and the state continues to evolve before the measurement is performed. This unfortunately means that it is unlikely for such devices implementing such a measurement protocol to be viable quantum thermal samples for generic quantum Hamiltonians.

Improved Boltzmann machines with error corrected quantum annealing, arXiv:1910.01283, Quantum Science and Technology, 5(4):045010 (2020)

In this paper, Richard Li, Daniel Lidar and I combine two previous ideas, quantum annealing correction (QAC) and training of a Boltzmann machine on a D-Wave quantum annealing processor. Even in the ideal scenario where the processor returns thermal samples, the temperature of the distribution is ultimately limited by the operating temperature of the device. Note that in the classical case, the temperature can be held fixed because the energy scale of the Hamiltonian can be rescaled arbitrarily, which is not something we can do on a physical device. By using multiple physical qubits, QAC can effectively raise the energy scale of the Ising Hamiltonian, which hopefully means we can achieve lower effective temperature. Unfortunately, we find there is typically a trade-off because as we couple more qubits together, it becomes harder for the system to appear thermal-like. This is ultimately not surprising since QAC does not encode the transverse field Hamiltonian, so flipping multiple qubits coherently is expected to become exponentially harder.

Validating a Two Qubit Non-Stoquastic Hamiltonian in Quantum Annealing, arXiv:1909.06333, Phys. Rev. A 101, 012310 (2020)

In this paper, I propose a 2-qubit protocol to validate a tunable antiferromagnetic XX coupler in quantum annealing. The presence of such a term in the Hamiltonian can make it non-stoquastic. From previous work, we know that such terms can lead to improvements in the performance of quantum annealing for certain problem classes. The challenge is to provide a clear 'signature’ within the constraints of quantum annealing. This means that the anneal has to end on a Hamiltonian that is diagonal in the computational basis, and only measurements at the end of the anneal in the computational basis are allowed. By tuning the strength of the antiferromagnetic XX coupler, we can change the instantaneous ground state near the end of the anneal from symmetric to antisymmetric in the swapping of the two qubits. This change can then be detected if the ground state at the end of the anneal is degenerate and includes states that are symmetric.

Permutation Matrix Representation Quantum Monte Carlo, arXiv:1908.03740, Journal of Statistical Mechanics: Theory and Experiment 2020, 073105 (2020)

In this paper, Lalit Gupta, Itay Hen, and I generalized the off-diagonal expansion (ODE) quantum Monte Carlo method introduced here to allow for a much more general and more unified approach to dealing with any type of (stoquastic) Hamiltonian. The approach allows for the treatment of many-body off-diagonal operators on arbitrary connectivity graphs. The key development is the use of permutation matrices to describe the action of the off-diagonal operators, for which we can define universal Monte Carlo updates.