arXiv subjects

James Ostrowski

Publications and source records attributed to James Ostrowski.

At least 19 recordsLinked to original sources

arXiv2026

Explicit Block Encoding of Difference-of-Gaussian Operators on a Periodic Grid

The Difference-of-Gaussian (DoG) is a widely used operator across applications, including image processing (feature and edge detection), quantum machine learning, and finite-difference methods (approximations of the Laplacian-of-Gaussian). In this paper, we construct an explicit quantum block encoding of the DoG operator on a periodic grid, exploiting its natural probabilistic structure. The central observation is that the DoG admits a natural decomposition to two normalized Gaussian distributions, each preparable by explicit and efficient circuits, with the negation encoded using a single Pauli-$Z$ gate on a branch-indicator qubit. This enables the operator's block encoding to be directly mapped to the Linear Combination of Unitaries framework without requiring signed amplitude loading, quantum random-access memory, or any other black-box oracles. The proposed method achieves a constant subnormalization factor $\lambda = 2$ independent of the grid size $N$, the spatial dimension $D$, and the stencil width. Additionally, we show that the DoG operator is diagonalized by the discrete Fourier basis, which allows us to derive an exact closed-form expression for the block-encoding success probability in terms of the input signal's power spectrum, weighted by the operator's transfer function. Finally, we prove that the expression reduces to $O(h^4)$ scaling with respect to grid spacing $h$ as the periodic grid becomes finer. This implementation provides an explicit construction method for a tunable, wide-stencil bandpass filter whose frequency response is controlled by two Gaussian scale parameters.

quant-ph↗

arXiv2025

Partitioned-Constraint QAOA (PC-QAOA): Structural State Preparation and Penalty Enforcement for Quantum Optimization

Constrained combinatorial optimization remains challenging for quantum algorithms because feasibility must be explicitly enforced, typically through penalty terms or problem-specific mixers. We introduce Partitioned-Constraint QAOA (PC-QAOA), which partitions constraints into those enforced structurally and those enforced energetically. Structural constraints are handled via feasible-state preparation and a Grover mixer that preserves feasibility, while the remaining constraints are enforced through penalties. We show that constraints with disjoint support can be prepared in parallel with little error accumulation. We identify broad classes of constraints (including cardinality, assignment, and flow conservation) that admit efficient structural enforcement, and introduce a variational gadget construction that extends this approach to arbitrary low-support constraints. Across 413 completed instances spanning multiple constraint families, PC-QAOA substantially improves feasibility and solution quality at shallow depth relative to penalty-based QAOA, demonstrating the value of partial structural enforcement.

quant-ph↗

arXiv2025

A mixed-integer program for circuit execution time minimization with precedence constraints

We present a mixed-integer programming (MIP) model for scheduling quantum circuits to minimize execution time. Our approach maximizes parallelism by allowing non-overlapping gates (those acting on distinct qubits) to execute simultaneously. Our methods apply to general circuits with precedence constraints. First, we derive closed-formulas for the execution time of circuits generated by ma-QAOA on star graphs for a layered, greedy, and MIP schedules. We then compare the MIP schedule against layered and greedy scheduling approaches on the circuits generated by ma-QAOA for solving the MaxCut problem on all non-isomorphic connected graphs with 3-7 vertices. These experiments demonstrate that the MIP scheduler consistently results in shorter circuit execution times than greedy and layered approaches, with up to 24\% savings.

quant-ph↗

arXiv2024

An angle rounding parameter initialization technique for ma-QAOA

The multi-angle quantum approximate optimization algorithm (ma-QAOA) is a recently introduced algorithm that gives at least the same approximation ratio as the quantum approximate optimization algorithm (QAOA) and, in most cases, gives a significantly higher approximation ratio than QAOA. One drawback to ma-QAOA is that it uses significantly more classical parameters than QAOA, so the classical optimization component more complex. In this paper, we motivate a new parameter initialization strategy in which angles are initially randomly set to multiples of $\pi/8$ between $-\pi$ and $\pi$ and this vector is used to seed one round of BFGS. We find that this parameter initialization strategy gives average approximation ratios of $0.900$, $0.982$, and $0.997$ for $p = 1, 2, 3$ layers of ma-QAOA. This is comparable to the average approximation ratios of ma-QAOA where the optimal parameters are found using BFGS with 1 random starting seed, which are $0.900$, $0.982$, and $0.996$. We also test another parameter initialization strategy in which angles corresponding to maximal degree vertices in the graph are set to 0 while all other are randomly initialized to random multiples of $\pi/8$. Using this strategy, the average approximation ratios are $0.897$, $0.984$, and $0.997$.

quant-ph↗

arXiv2024

Quantum approximate optimization algorithm with random and subgraph phase operators

The quantum approximate optimization algorithm (QAOA) is a promising quantum algorithm that can be used to approximately solve combinatorial optimization problems. The usual QAOA ansatz consists of an alternating application of the cost and mixer Hamiltonians. In this work, we study how using Hamiltonians other than the usual cost Hamiltonian, dubbed custom phase operators, can affect the performance of QAOA. We derive an expected value formula for QAOA with custom phase operators at $p = 1$ and show numerically that some of these custom phase operators can achieve higher approximation ratio than the original algorithm implementation. Out of all the graphs tested at $p=1$, 0.036\% of the random custom phase operators, 75.9\% of the subgraph custom phase operators, 95.1\% of the triangle-removed custom phase operators, and 93.9\% of the maximal degree edge-removed custom phase operators have a higher approximation ratio than the original QAOA implementation. Furthermore, we numerically simulate these phase operators for $p=2$ and $p=3$ levels of QAOA and find that there exist a large number of subgraph, triangle-removed, and maximal degree edge-removed custom phase operators that have a higher approximation ratio than QAOA at the same depth. These findings open up the question of whether better phase operators can be designed to further improve the performance of QAOA.

quant-ph↗

arXiv2024

On the optimal objective value of random linear programs

We consider the problem of maximizing $\langle c,x \rangle$ subject to the constraints $Ax \leq \mathbf{1}$, where $x\in R^n$, $A$ is an $m\times n$ matrix with mutually independent centered subgaussian entries of unit variance, and $c$ is a cost vector of unit Euclidean length. In the asymptotic regime $n\to\infty$, $\frac{m}{n}\to\infty$, and under some mild assumptions on $c$, we prove that the optimal objective value $z^*$ of the linear program satisfies $$ \lim\limits_{n\to\infty}\sqrt{2\log(m/n)}\,z^*= 1\quad \mbox{almost surely}. $$ We provide numerical experiments as supporting data for the theoretical predictions. Further, we carry out numerical studies of the limiting distribution and the standard deviation of $z^*$.

math.PR↗

arXiv2023

Combinatorial optimization with quantum imaginary time evolution

We use Quantum Imaginary Time Evolution (QITE) to solve polynomial unconstrained binary optimization (PUBO) problems. We show that a linear Ansatz yields good results for a wide range of PUBO problems, often outperforming standard classical methods, such as the Goemans-Williamson (GW) algorithm. We obtain numerical results for the Low Autocorrelation Binary Sequences (LABS) and weighted MaxCut combinatorial optimization problems, thus extending an earlier demonstration of successful application of QITE on MaxCut for unweighted graphs. We find the performance of QITE on the LABS problem with a separable Ansatz comparable with p=10 QAOA, and do not see a significant advantage with an entangling Ansatz. On weighted MaxCut, QITE with a separable Ansatz often outperforms the GW algorithm on graphs up to 150 vertices.

quant-ph↗

arXiv2023

Graph decomposition techniques for solving combinatorial optimization problems with variational quantum algorithms

The quantum approximate optimization algorithm (QAOA) has the potential to approximately solve complex combinatorial optimization problems in polynomial time. However, current noisy quantum devices cannot solve large problems due to hardware constraints. In this work, we develop an algorithm that decomposes the QAOA input problem graph into a smaller problem and solves MaxCut using QAOA on the reduced graph. The algorithm requires a subroutine that can be classical or quantum--in this work, we implement the algorithm twice on each graph. One implementation uses the classical solver Gurobi in the subroutine and the other uses QAOA. We solve these reduced problems with QAOA. On average, the reduced problems require only approximately 1/10 of the number of vertices than the original MaxCut instances. Furthermore, the average approximation ratio of the original MaxCut problems is 0.75, while the approximation ratios of the decomposed graphs are on average of 0.96 for both Gurobi and QAOA. With this decomposition, we are able to measure optimal solutions for ten 100-vertex graphs by running single-layer QAOA circuits on the Quantinuum trapped-ion quantum computer H1-1, sampling each circuit only 500 times. This approach is best suited for sparse, particularly $k$-regular graphs, as $k$-regular graphs on $n$ vertices can be decomposed into a graph with at most $\frac{nk}{k+1}$ vertices in polynomial time. Further reductions can be obtained with a potential trade-off in computational time. While this paper applies the decomposition method to the MaxCut problem, it can be applied to more general classes of combinatorial optimization problems.

quant-ph↗

arXiv2023

Densely Connected $G$-invariant Deep Neural Networks with Signed Permutation Representations

We introduce and investigate, for finite groups $G$, $G$-invariant deep neural network ($G$-DNN) architectures with ReLU activation that are densely connected-- i.e., include all possible skip connections. In contrast to other $G$-invariant architectures in the literature, the preactivations of the$G$-DNNs presented here are able to transform by \emph{signed} permutation representations (signed perm-reps) of $G$. Moreover, the individual layers of the $G$-DNNs are not required to be $G$-equivariant; instead, the preactivations are constrained to be $G$-equivariant functions of the network input in a way that couples weights across all layers. The result is a richer family of $G$-invariant architectures never seen previously. We derive an efficient implementation of $G$-DNNs after a reparameterization of weights, as well as necessary and sufficient conditions for an architecture to be ``admissible''-- i.e., nondegenerate and inequivalent to smaller architectures. We include code that allows a user to build a $G$-DNN interactively layer-by-layer, with the final architecture guaranteed to be admissible. We show that there are far more admissible $G$-DNN architectures than those accessible with the ``concatenated ReLU'' activation function from the literature. Finally, we apply $G$-DNNs to two example problems -- (1) multiplication in $\{-1, 1\}$ (with theoretical guarantees) and (2) 3D object classification -- % finding that the inclusion of signed perm-reps significantly boosts predictive performance compared to baselines with only ordinary (i.e., unsigned) perm-reps.

cs.LG↗

arXiv2022

Approximate Boltzmann Distributions in Quantum Approximate Optimization

Approaches to compute or estimate the output probability distributions from the quantum approximate optimization algorithm (QAOA) are needed to assess the likelihood it will obtain a quantum computational advantage. We analyze output from QAOA circuits solving 7,200 random MaxCut instances, with $n=14-23$ qubits and depth parameter $p \leq 12$, and find that the average basis state probabilities follow approximate Boltzmann distributions: The average probabilities scale exponentially with their energy (cut value), with a peak at the optimal solution. We describe the rate of exponential scaling or "effective temperature" in terms of a series with a leading order term $T \sim C_\mathrm{min}/n\sqrt{p}$, with $C_\mathrm{min}$ the optimal solution energy. Using this scaling we generate approximate output distributions with up to 38 qubits and find these give accurate accounts of important performance metrics in cases we can simulate exactly.

quant-ph↗

arXiv2022

A Classification of $G$-invariant Shallow Neural Networks

When trying to fit a deep neural network (DNN) to a $G$-invariant target function with $G$ a group, it only makes sense to constrain the DNN to be $G$-invariant as well. However, there can be many different ways to do this, thus raising the problem of ``$G$-invariant neural architecture design'': What is the optimal $G$-invariant architecture for a given problem? Before we can consider the optimization problem itself, we must understand the search space, the architectures in it, and how they relate to one another. In this paper, we take a first step towards this goal; we prove a theorem that gives a classification of all $G$-invariant single-hidden-layer or ``shallow'' neural network ($G$-SNN) architectures with ReLU activation for any finite orthogonal group $G$, and we prove a second theorem that characterizes the inclusion maps or ``network morphisms'' between the architectures that can be leveraged during neural architecture search (NAS). The proof is based on a correspondence of every $G$-SNN to a signed permutation representation of $G$ acting on the hidden neurons; the classification is equivalently given in terms of the first cohomology classes of $G$, thus admitting a topological interpretation. The $G$-SNN architectures corresponding to nontrivial cohomology classes have, to our knowledge, never been explicitly identified in the literature previously. Using a code implementation, we enumerate the $G$-SNN architectures for some example groups $G$ and visualize their structure. Finally, we prove that architectures corresponding to inequivalent cohomology classes coincide in function space only when their weight matrices are zero, and we discuss the implications of this for NAS.

cs.LG↗

arXiv2022

A Group-Equivariant Autoencoder for Identifying Spontaneously Broken Symmetries

We introduce the group-equivariant autoencoder (GE-autoencoder) -- a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE-autoencoder such that the encoder learns an order parameter invariant to these ``never-broken'' symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE-autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE-autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE-autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time-efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Finally, we describe various key implementation details, including a new method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.

cond-mat.stat-mech↗

arXiv2022

Solving MaxCut with Quantum Imaginary Time Evolution

We introduce a method to solve the MaxCut problem efficiently based on quantum imaginary time evolution (QITE). We employ a linear Ansatz for unitary updates and an initial state involving no entanglement, as well as an imaginary-time-dependent Hamiltonian interpolating between a given graph and a subgraph with two edges excised. We apply the method to thousands of randomly selected graphs with up to fifty vertices. We show that our algorithm exhibits a 93% and above performance converging to the maximum solution of the MaxCut problem for all considered graphs. Our results compare favorably with the performance of classical algorithms, such as the greedy and Goemans-Williamson algorithms. We also discuss the overlap of the final state of the QITE algorithm with the ground state as a performance metric, which is a quantum feature not shared by other classical algorithms. This metric can be improved by introducing higher-order Ansaetze and entangled initial states.

quant-ph↗

arXiv2022

Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut

Finding high-quality parameters is a central obstacle to using the quantum approximate optimization algorithm (QAOA). Previous work partially addresses this issue for QAOA on unweighted MaxCut problems by leveraging similarities in the objective landscape among different problem instances. However, we show that the more general weighted MaxCut problem has significantly modified objective landscapes, with a proliferation of poor local optima. Our main contribution is a simple rescaling scheme that overcomes these deleterious effects of weights. We show that for a given QAOA depth, a single "typical" vector of QAOA parameters can be successfully transferred to weighted MaxCut instances. This transfer leads to a median decrease in the approximation ratio of only 2.0 percentage points relative to a considerably more expensive direct optimization on a dataset of 34,701 instances with up to 20 nodes and multiple weight distributions. This decrease can be reduced to 1.2 percentage points at the cost of only 10 additional QAOA circuit evaluations with parameters sampled from a pretrained metadistribution, or the transferred parameters can be used as a starting point for a single local optimization run to obtain approximation ratios equivalent to those achieved by exhaustive optimization in $96.35\%$ of our cases.

quant-ph↗

arXiv2022

Scaling Quantum Approximate Optimization on Near-term Hardware

The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.

quant-ph↗

arXiv2021

Multi-angle Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we introduce a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time. This new ansatz gives a 33\% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.

quant-ph↗

arXiv2021

Globally optimizing QAOA circuit depth for constrained optimization problems

We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.

quant-ph↗

arXiv2021

Empirical performance bounds for quantum approximate optimization

The quantum approximate optimization algorithm (QAOA) is a variational method for noisy, intermediate-scale quantum computers to solve combinatorial optimization problems. Quantifying performance bounds with respect to specific problem instances provides insight into when QAOA may be viable for solving real-world applications. Here, we solve every instance of MaxCut on non-isomorphic unweighted graphs with nine or fewer vertices by numerically simulating the pure-state dynamics of QAOA. Testing up to three layers of QAOA depth, we find that distributions of the approximation ratio narrow with increasing depth while the probability of recovering the maximum cut generally broadens. We find QAOA exceeds the Goemans-Williamson approximation ratio bound for most graphs. We also identify consistent patterns within the ensemble of optimized variational circuit parameters that offer highly efficient heuristics for solving MaxCut with QAOA. The resulting data set is presented as a benchmark for establishing empirical bounds on QAOA performance that may be used to test on-going experimental realizations.

quant-ph↗