Partial Syndrome Measurement for Hypergraph Product Codes

Hypergraph product codes are a promising avenue to achieving fault-tolerant quantum computation with constant overhead. When embedding these and other constant-rate qLDPC codes into 2D, a significant number of nonlocal connections are required, posing difficulties for some quantum computing architectures. In this work, we introduce a fault-tolerance scheme that aims to alleviate the effects of implementing this nonlocality by measuring generators acting on spatially distant qubits less frequently than those which do not. We investigate the performance of a simplified version of this scheme, where the measured generators are randomly selected. When applied to hypergraph product codes and a modified small-set-flip decoding algorithm, we prove that for a sufficiently high percentage of generators being measured, a threshold still exists. We also find numerical evidence that the logical error rate is exponentially suppressed even when a large constant fraction of generators are not measured.Comment: 10 pages, 4 figure

Learning crystal field parameters using convolutional neural networks

We present a deep machine learning algorithm to extract crystal field (CF) Stevens parameters from thermodynamic data of rare-earth magnetic materials. The algorithm employs a two-dimensional convolutional neural network (CNN) that is trained on magnetization, magnetic susceptibility and specific heat data that is calculated theoretically within the single-ion approximation and further processed using a standard wavelet transformation. We apply the method to crystal fields of cubic, hexagonal and tetragonal symmetry and for both integer and half-integer total angular momentum values $J$ of the ground state multiplet. We evaluate its performance on both theoretically generated synthetic and previously published experimental data on CeAgSb$_2$, PrAgSb$_2$ and PrMg$_2$Cu$_9$, and find that it can reliably and accurately extract the CF parameters for all site symmetries and values of $J$ considered. This demonstrates that CNNs provide an unbiased approach to extracting CF parameters that avoids tedious multi-parameter fitting procedures.Comment: 19 pages, 9 figure

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

We perform a systematic study of preparing ground states of correlated multi-orbital impurity models using variational quantum eigensolvers (VQEs). We consider both fixed and adaptive wavefunction ans\"atze and analyze the resulting gate depths and the performance with and without noise. For the adaptive procedure, we develop an operator pool consisting of pairwise commutators of Hamiltonian terms that allows for a fair comparison between the adaptive and fixed Hamiltonian variational ansatz. Using noiseless statevector simulations, we find that the most compact ans\"atze are obtained in an atomic orbital representation and using parity encoding. Focusing on the adaptive algorithms, which yield the circuits with the least number of CNOTs, we then show that in the presence of sampling noise, high-fidelity state preparation can still be achieved with the Hamiltonian commutator pool. By utilizing Hamiltonian integral factorization and a noise resilient optimizer, we show that this approach requires only a modest number of about $2^{12}$ shots per measurement circuit. We discover a dichotomy of the operator pool complexity in the presence of sampling noise, where a small pool size reduces the adaptive overhead but a larger pool size accelerates convergence to the ground state. When considering realistic gate noise in addition, we observe that the variable optimization can still be performed as long as the two-qubit gate error lies below $10^{-3}$, which is close but below current hardware levels. Finally, we measure the ground state energy of the $e_g$ model on IBM and Quantinuum quantum hardware using the converged adaptive ansatz. We perform a systematic error mitigation analysis on the IBM results and obtain a relative error of 0.7\% using symmetry-based postselection and zero-noise extrapolation (ZNE).Comment: 19 pages, 9 figure

Efficient Step-Merged Quantum Imaginary Time Evolution Algorithm for Quantum Chemistry

We develop a resource-efficient step-merged quantum imaginary time evolution approach (smQITE) to solve for the ground state of a Hamiltonian on quantum computers. This heuristic method features a fixed shallow quantum circuit depth along the state evolution path. We use this algorithm to determine the binding energy curves of a set of molecules, including H2, H4, H6, LiH, HF, H2O, and BeH2, and find highly accurate results. The required quantum resources of smQITE calculations can be further reduced by adopting the circuit form of the variational quantum eigensolver (VQE) technique, such as the unitary coupled cluster ansatz. We demonstrate that smQITE achieves a similar computational accuracy as VQE at the same fixed-circuit ansatz, without requiring a generally complicated high-dimensional nonconvex optimization. Finally, smQITE calculations are carried out on Rigetti quantum processing units, demonstrating that the approach is readily applicable on current noisy intermediate-scale quantum devices

Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations

Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled cluster ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit. With these strategies, VQE calculations on a noiseless quantum simulator achieve energies within a few meVs of those obtained with the full UCCSD ansatz for $\mathrm{H}_4$ square, $\mathrm{H}_4$ chain and $\mathrm{H}_6$ hexagon molecules; while the number of controlled-NOT (CNOT) gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using $\mathrm{H}_4$ square and $\mathrm{H}_4$ chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational ansatz

Quantum dynamics simulations beyond the coherence time on NISQ hardware by variational Trotter compression

We demonstrate a post-quench dynamics simulation of a Heisenberg model on present-day IBM quantum hardware that extends beyond the coherence time of the device. This is achieved using a hybrid quantum-classical algorithm that propagates a state using Trotter evolution and then performs a classical optimization that effectively compresses the time-evolved state into a variational form. When iterated, this procedure enables simulations to arbitrary times with an error controlled by the compression fidelity and a fixed Trotter step size. We show how to measure the required cost function, the overlap between the time-evolved and variational states, on present-day hardware, making use of several error mitigation methods. In addition to carrying out simulations on real hardware, we investigate the performance and scaling behavior of the algorithm with noiseless and noisy classical simulations. We find the main bottleneck in going to larger system sizes to be the difficulty of carrying out the optimization of the noisy cost function.This is a pre-print of the article Berthusen, Noah F., Thaís V. Trevisan, Thomas Iadecola, and Peter P. Orth. "Quantum dynamics simulations beyond the coherence time on NISQ hardware by variational Trotter compression." arXiv preprint arXiv:2112.12654 (2021). DOI: 10.48550/arXiv.2112.12654. Copyright 2022 The Authors. Posted with permission

Learning crystal field parameters using convolutional neural networks

We present a deep machine learning algorithm to extract crystal field (CF) Stevens parameters from thermodynamic data of rare-earth magnetic materials. The algorithm employs a two-dimensional convolutional neural network (CNN) that is trained on magnetization, magnetic susceptibility and specific heat data that is calculated theoretically within the single-ion approximation and further processed using a standard wavelet transformation. We apply the method to crystal fields of cubic, hexagonal and tetragonal symmetry and for both integer and half-integer total angular momentum values J of the ground state multiplet. We evaluate its performance on both theoretically generated synthetic and previously published experimental data on CeAgSb 2 , PrAgSb 2 and PrMg 2 Cu 9 , and find that it can reliably and accurately extract the CF parameters for all site symmetries and values of J considered. This demonstrates that CNNs provide an unbiased approach to extracting CF parameters that avoids tedious multi-parameter fitting procedures.</p

Quantum dynamics simulations beyond the coherence time on NISQ hardware by variational Trotter compression

We demonstrate a post-quench dynamics simulation of a Heisenberg model on present-day IBM quantum hardware that extends beyond the coherence time of the device. This is achieved using a hybrid quantum-classical algorithm that propagates a state using Trotter evolution and then performs a classical optimization that effectively compresses the time-evolved state into a variational form. When iterated, this procedure enables simulations to arbitrary times with an error controlled by the compression fidelity and a fixed Trotter step size. We show how to measure the required cost function, the overlap between the time-evolved and variational states, on present-day hardware, making use of several error mitigation methods. In addition to carrying out simulations on real hardware, we investigate the performance and scaling behavior of the algorithm with noiseless and noisy classical simulations. We find the main bottleneck in going to larger system sizes to be the difficulty of carrying out the optimization of the noisy cost function.Comment: 11 pages, 9 figure

Differentiation of Myositis-Induced Models of Bacterial Infection and Inflammation with T2-Weighted, CEST, and DCE-MRI

We used T2 relaxation, chemical exchange saturation transfer (CEST), and dynamic contrast-enhanced (DCE) magnetic resonance imaging (MRI) to assess whether bacterial infection can be differentiated from inflammation in a myositis-induced mouse model. We measured the T2 relaxation time constants, %CEST at 5 saturation frequencies, and area under the curve (AUC) from DCE-MRI after maltose injection from infected, inflamed, and normal muscle tissue models. We applied principal component analysis (PCA) to reduce dimensionality of entire CEST spectra and DCE signal evolutions, which were analyzed using standard classification methods. We extracted features from dimensional reduction as predictors for machine learning classifier algorithms. Normal, inflamed, and infected tissues were evaluated with H&amp;E and gram-staining histological studies, and bacterial-burden studies. The T2 relaxation time constants and AUC of DCE-MRI after injection of maltose differentiated infected, inflamed, and normal tissues. %CEST amplitudes at −1.6 and −3.5 ppm differentiated infected tissues from other tissues, but these did not differentiate inflamed tissue from normal tissue. %CEST amplitudes at 3.5, 3.0, and 2.5 ppm, AUC of DCE-MRI for shorter time periods, and relative Ktrans and kep values from DCE-MRI could not differentiate tissues. PCA and machine learning of CEST-MRI and DCE-MRI did not improve tissue classifications relative to traditional analysis methods. Similarly, PCA and machine learning did not further improve tissue classifications relative to T2 MRI. Therefore, future MRI studies of infection models should focus on T2-weighted MRI and analysis of T2 relaxation times

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

Hybrid quantum-classical embedding methods for correlated materials simulations provide a path towards potential quantum advantage. However, the required quantum resources arising from the multi-band nature of d and f electron materials remain largely unexplored. Here we compare the performance of different variational quantum eigensolvers in ground state preparation for interacting multi-orbital embedding impurity models, which is the computationally most demanding step in quantum embedding theories. Focusing on adaptive algorithms and models with 8 spin-orbitals, we show that state preparation with fidelities better than 99.9% can be achieved using about 214 shots per measurement circuit. When including gate noise, we observe that parameter optimizations can still be performed if the two-qubit gate error lies below 10−3, which is slightly smaller than current hardware levels. Finally, we measure the ground state energy on IBM and Quantinuum hardware using a converged adaptive ansatz and obtain a relative error of 0.7%.This article is published as Mukherjee, Anirban, Noah F. Berthusen, João C. Getelina, Peter P. Orth, and Yong-Xin Yao. "Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models." Communications Physics 6, no. 1 (2023): 4. DOI: 10.1038/s42005-022-01089-6. Copyright 2023 The Author(s). Attribution 4.0 International (CC BY 4.0). Posted with permission. DOE Contract Number(s): AC02-07CH11358; AC05-00OR2272