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Mitchell A. Thornton

Publications and source records attributed to Mitchell A. Thornton.

At least 19 recordsLinked to original sources
arXiv2026

A Version Space Approach for Digital Circuit Analysis

Many questions about a digital circuit take the same form. A hidden object is consistent with a set of observations, and one wants to know how many remain consistent and which observation to make next. The set of surviving candidates is the version space, and its size, on a logarithmic scale, measures how much the observations have settled. This paper develops the version-space view as one method and applies it to two circuit-analysis problems usually treated as unrelated. The first is probabilistic combinational equivalence checking, where the candidates are Boolean functions and the observations are modified-Haar spectral coefficients. A method proposed in 2002 posed this counting problem and solved only two special cases, leaving the general case an enumeration exponential in the number of observations. We close it. A reparameterization onto block sums turns the dependence among nested coefficients into locality, a sum--product recursion counts the surviving functions exactly in time polynomial in the truth-table size, closed forms follow for a single coefficient, a coefficient pair, and every ancestor-closed set, and the error of the independence approximation the 2002 work resorted to equals a computable lattice index. Every formula is checked against exhaustive enumeration and reproduces the 2002 tables. The second application is key counting for logic-locked netlists, where the candidates are keys and the observations are oracle responses. The same recursion, run over the gate-level factor graph, computes the number of keys still consistent with a set of queries; across seventy instances of the TrustHub obfuscation release the surviving entropy falls below the advertised key length every time. The two applications are one method: a witness supplies observations, each removes candidates, and the version space is counted exactly.

cs.CR↗
arXiv2026

RENESIS: Energy-Aware Synthesis of Adiabatic Logic from Irreversible Netlists

We describe Renesis, an automated synthesis tool that accepts an ordinary irreversible netlist and produces a verified, technology-mapped energy-recovery (adiabatic) circuit, using energy rather than area or delay as the optimization criterion. Renesis models the netlist with a vector-space formulation that expresses simulation and justification sweeps as forward and reverse traversals whose cost is linear in the number of circuit components. The traversals populate ledgers with data tags that characterize switching, erasure, and observability information at their natural R\'enyi orders. The output is a logically reversible circuit mapped to one of eight energy-recovery families, with the associated parameters reported. Reversibility is treated here as a circuit-level requirement rather than a thermodynamic one. When an adiabatic gate erases information the penalty is not $k_B T \ln 2$ but a full non-adiabatic $CV^2$ discharge, which is comparable to the switching energy the circuit style exists to recover. Every synthesis transformation is equivalence-checked, and it must improve one of two reported cost tables, one uncapped and one after a series-realizability bound, while worsening neither before it is accepted. Across a twenty-circuit development set, optional re-synthesis passes improve fourteen circuits. On a held-out set of twenty circuits, fifteen of nineteen are improved, with a best-arm median of $0.91$ of the default energy. A certified optimality-gap program computes the distance between the synthesized circuits and the provable floor of the tool's own search space. A device-level SPICE deck reproduces the tool's per-cycle energy figures on the reference family. The tool, the benchmark netlists, the validation procedure, and the run records behind every reported number are released as open source.

cs.AR↗
arXiv2026

Matched Generators for the Karhunen--Lo\`eve Transform: A Double-Commutator Eigenvalue Theory

The Karhunen--Lo\`eve transform (KLT) diagonalizes the covariance of a second-order process and is optimal for mean-square truncation. Which classical transform it reduces to is governed by the symmetry commutant of the covariance: when the kernel commutes with a group action, the KLT eigenfunctions are the irreducible representation functions of that group, recovering the Fourier, cosine, Mellin, and spherical-harmonic systems. We study the inverse question. Given a covariance $R$ and a finite-dimensional space of candidate generators, the generator nearest to commuting with $R$, the minimizer of $\delta(A,R)=\|[R,A]\|_F/(\|R\|_F\|A\|_F)$, is the smallest-eigenvalue solution of a double-commutator eigenvalue problem $\mathrm{ad}_R^2(A^\ast)=\lambda A^\ast$, a Hermitian generalized eigenvalue problem of size the number of generators, independent of dimension. The framework recovers hidden transforms as well as classical ones: a variational characterization turns the existence of a commuting generator into a spectral condition, and a tridiagonal commutant-uniqueness result yields the prolate spheroidal, cosine, and discrete orthogonal-polynomial bases as exact recoveries, with matrix-valued extensions, and produces a continuum of transforms interpolating between and beyond the classical families. When symmetry is approximate, the coding penalty of the symmetry-adapted blockwise transform equals the multi-information among the sectors, an exact threshold between the fixed and data-driven transforms. We further give a graph-automorphism characterization of permutation structure, a sequential deflation for non-Abelian symmetry, and stability bounds under estimation error. As an application, the KLT of a two-paradigm covariance is synthesized from its two known generators, without forming the mixed covariance, reaching the full-data transform's compaction from few observations.

eess.SP↗
arXiv2026

Blind Symmetry Matching in Quantum States with Application to Shot-Count Reduction

Measuring a quantum computation in a basis adapted to a symmetry it carries reduces the repeated measurements, commonly referred to as ``shots'', needed to read a statistical answer. Detecting the symmetry a quantum state carries has many uses: certifying a claimed symmetry, identifying a conserved-charge sector, flagging symmetry-breaking as an error signature, and selecting a compression or readout basis; shot-count reduction is developed here as one exemplary case. Existing methods assume the symmetry is known in advance; we remove that assumption. When it is unknown, the carried symmetry is discovered from the data by a symmetry test that scores candidate groups, and the largest passing group is exploited as the measurement basis. We state the pipeline precisely, prove the selection rule is unbiased, and charge discovery in full. Two conditions are treated, both detected by the same score with a different projection: a weak condition, commutation with the representation, and a strong condition, confinement to a single charge sector, the distinction drawn in the quantum-reference-frame literature. A single circuit, a controlled twirl followed by a SWAP test, discovers both: discarding the group register tests the weak condition, post-selecting it the strong one. The framework is general over finite groups, with cyclic (Fourier), dihedral, and symmetric-group (Schur-Weyl) examples; strong confinement to the symmetric, or Dicke, subspace is an exponential reduction. Seeded demonstrations show the loop wins net of discovery: weak matching on momentum readout reduces shots by a factor widening from ten to several thousand, and strong matching on a two-system target by a further factor of the subsystem size. Blind symmetry matching is a practical primitive for the common case where the matched basis cannot be written down in advance.

quant-ph↗
arXiv2026

Kernel Density Estimation by Spectral Decomposition: Data-Driven Tapering and Superposition

Kernel density estimation depends largely on one choice, the smoothing bandwidth. We treat bandwidth selection and density estimation in the characteristic-function domain, where the cyclic group-averaged covariance of the binned data has the squared empirical characteristic function as its spectrum: the true characteristic function sits over a sampling-noise floor of $1/n$, and the bandwidth is the spectral cutoff where the two meet. Several methods follow. An automatic selector strips the floor and minimizes a frequency-domain error criterion, matching the rule of thumb on smooth densities and approaching the best fixed bandwidth on multimodal ones. An adaptive estimator generalizes the fixed kernel to the per-frequency optimal Wiener taper, matching or surpassing the best fixed bandwidth on most standard densities, including sharply peaked and comb-like cases where fixed bandwidths fail; deconvolution under known measurement error follows in the same domain. Because the Wiener estimator resolves sharp structure but does not fit smooth bases as economically as a mixture, a Gaussian mixture is combined with it two ways, a piecewise partition and a superposition of a smooth base and a band-limited residual, the default. A data-driven floor read from the spectrum replaces the assumed $1/n$ floor and stays robust on heaped and rounded data. On the Marron-Wand benchmark scored by exact integrated squared error, the advantage emerges with sample size, a bias-variance tradeoff: the spectral estimators carry low bias but pay in variance, so a corrected Botev plug-in leads at $n=100$ while the Wiener filter and superposition take the top two ranks at $n=5000$. The methods are validated on six real datasets (CRSP returns, NHANES self-reports, CMS dimuon and SDSS spectra, a random-beacon stream, and UNSW-NB15 traffic) and on a synthetic-data quality check. All experiments are reproducible.

stat.ME↗
arXiv2026

Symmetry-Aware Convex Shrinkage for High-Dimensional Covariance Estimation

We develop a class of data-adaptive shrinkage estimators for high-dimensional covariance estimation in which the shrinkage target is a Reynolds projection of the sample covariance under a finite symmetry group selected from a candidate library by held-out predictive performance. The class generalizes the convex shrinkage estimator of Ledoit and Wolf by replacing the scalar-identity target with a structured target derived from a symmetry group when one is available, and generalizes the group-symmetric maximum-likelihood estimator of Shah and Chandrasekaran by combining structural targeting with adaptive convex shrinkage and by selecting the group from data rather than treating it as prespecified. A two-tier procedure performs the group selection: a universal per-candidate evaluation based on held-out negative log-likelihood, optionally preceded by a domain-specific step that constructs the candidate library from structural priors. We establish a finite-sample regret bound for the held-out calibration of the convex combination weight, an oracle inequality for the data-driven group selection, and a quantitative sufficient-match condition under which the proposed estimator dominates Ledoit-Wolf shrinkage in Frobenius mean-squared error. The procedure is illustrated on six real-data problems spanning finance (S&P~500 daily returns), climate (NOAA OISST sea-surface temperature anomalies), genomics (TCGA-BRCA gene expression), radio signal processing (RadioML 2018.A), astronomical imaging (Galaxy10 DECaLS), and natural image patches (CIFAR-10 with a CIFAR-10.1 distribution-shift companion). An empirical comparison is also made against the Bayesian permutation-symmetry estimator of Chojecki and colleagues. Outside the few-shot regime, where structural priors carry the most information per observation, Ledoit-Wolf shrinkage remains the appropriate baseline.

stat.ME↗
arXiv2026

Unification of Signal Transform Theory

We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Lo\`eve transform (KLT), and several others along with their continuous counterparts (Fourier transform, Fourier series, spherical harmonics, fractional Fourier transform) under one representation-theoretic principle: each is the eigenbasis of every covariance invariant under a specific finite or compact group, with columns constructed from the irreducible matrix elements of the group via the Peter-Weyl theorem. The unification rests on the Algebraic Diversity (AD) framework, which identifies the matched group of a covariance as the foundational object of second-order signal processing. The data-dependent KLT emerges as the trivial-matched-group limit; classical transforms emerge as the cyclic, dihedral, elementary Abelian, iterated wreath, and hybrid wreath cases, with composition rules for direct, wreath, and semidirect products. The dihedral case is split by action: the Hartley basis is the matched transform of the dihedral group on the $M$-cycle (symmetric circulants), while the DCT belongs to the dihedral group on the even-reflected extension, with the finite-window AR(1) covariance an approximation to that class. A polynomial-time algorithm, the DAD-CAD relaxation cast as a double-commutator generalized eigenvalue problem, discovers the matched group of any empirical covariance without expert judgment, with noise-aware variants via the commutativity residual $\delta$ and algebraic coloring index $\alpha$. The fractional Fourier transform is treated as the metaplectic $SO(2)$ case, and a structural principle relates matched group size inversely to transform resolution. Modern applications (massive-MIMO, graph neural networks, transformer attention, 3D vision, brain connectivity, single-cell genomics, quantum informatics) are sketched with their matched groups.

eess.SP↗
arXiv2026

Polynomial-Time Optimal Group Selection via the Double-Commutator Eigenvalue Problem

The algebraic diversity framework generalizes temporal averaging over multiple observations to algebraic group action on a single observation for second-order statistical estimation. The central open problem in this framework is $\textit{group selection}$: given an $M$-dimensional observation with unknown covariance structure, find the finite group whose spectral decomposition best matches the covariance. Naive enumeration of all subgroups of the symmetric group $S_M$ requires exponential time in $M$. We prove that this combinatorial problem reduces to a generalized eigenvalue problem derived from the double commutator of the covariance matrix, yielding a polynomial-time algorithm with complexity $O(d^2M^2 + d^3)$, where $d$ is the dimension of a generator basis. The minimum eigenvector of the double-commutator matrix directly constructs the optimal group generator in closed form, with no iterative optimization. The reduction is exact: the double-commutator minimum eigenvalue is zero if and only if the optimal generator lies in the span of the basis, and its magnitude provides a certifiable optimality gap when it does not. This problem does not appear in the standard catalogs of computational complexity (Garey and Johnson, 1979) and represents a new class linking group theory, matrix analysis, and statistical estimation. We establish connections to independent component analysis (JADE), structured matrix nearness problems, and simultaneous matrix diagonalization, and we show that the double-commutator formulation is the unique approach that is simultaneously polynomial-time, closed-form, and certifiable. We extend the framework to non-Abelian symmetry recovery via a Sequential GEVP with deflation, and add two identifiability theorems characterizing the commutant-lattice ambiguity and the dichotomy on whether $\mathrm{Aut}(\mathbf{R})$ recovers a generative subgroup or only a supergroup.

cs.LG↗
arXiv2026

Algebraic Diversity: Principles of a Group-Theoretic Approach to Signal Processing

We present principles of algebraic diversity (AD), a group-theoretic approach to signal processing exploiting signal symmetry to extract more information per observation, complementing classical methods that use temporal and spatial diversity. The transformations under which a signal's statistics are invariant form a matched group; this group determines the natural transform for analysis, and averaging an estimator over the group action reduces variance without requiring additional snapshots. The viewpoint is broadened in five directions beyond the single-observation measurement of a companion paper. Rank promotion admits AD on scalar data streams and identifies the law of large numbers as the trivial-group case of a $(G, L)$ continuum combining sample-count with group-orbit averaging. An eigentensor hierarchy handles signals with nested symmetry. A blind group-matching methodology identifies the matched group from data via a polynomial-time generalized eigenvalue problem on the unitary Lie algebra, placing the DFT, DCT, and Karhunen--Lo\`{e}ve transforms as distinguished points on a transform manifold. A cost-symmetry matching principle then extends AD from measurement to blind and adaptive signal processing generally; blind equalization is given as a detailed example, with the Constant Modulus Algorithm's residual phase ambiguity predicted analytically and matched within two degrees on 3GPP TDL multipath channels, and other blind problems in signal processing are mapped into the framework. Four theorems formalize a structural capacity $\kappa$, the R\'{e}nyi-2 analog of Shannon and von Neumann's R\'{e}nyi-1 entropies, quantifying how a signal's information is organized rather than how much information it contains. AD relationship to prior algebraic approaches including invariant estimation, minimax robust estimation, algebraic signal processing, and compressed sensing.

eess.SP↗
arXiv2026

Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions

We provide a computable criterion for selecting among Fourier, wavelet, and time-frequency analysis by extending the algebraic diversity (AD) framework to Lie groups acting on $L^2(\mathbb{R})$. To our knowledge, there is no other criterion that provides this selection capability. The group-averaged estimator generalizes from a finite sum over group elements to an integral with respect to Haar measure. A Continuous Replacement Theorem establishes signal-noise separation under equivariance and ergodicity conditions, with a noise operator $\mathcal{N}_G = C_ρ^{-2}$ determined by the Duflo-Moore operator that explains the frequency-dependent noise floor in wavelet analysis as a consequence of the affine group's non-unimodularity. A Unification Theorem shows that classical spectral analysis corresponds to the translation group, wavelet analysis to the affine group, time-frequency analysis to the Heisenberg-Weyl group, and spherical harmonics to SO(3). The commutativity residual $δ$, extended to Hilbert-Schmidt operator norms, provides a principled selection criterion among these groups. A double-commutator generalized eigenvalue problem solves the blind group matching problem in polynomial time. A Discretization Recovery Theorem establishes that all discrete AD results are sampling approximations to the continuous theory, with $\mathbb{Z}_M \to (\mathbb{R},+)$ as $M \to \infty$.

eess.SP↗
arXiv2026

Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

We establish that temporal averaging over multiple observations is the degenerate case of algebraic group action with the trivial group $G=\{e\}$. A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation. The Trivial Group Embedding Theorem proves that the sample covariance is the accumulation of trivial-group estimates, with variance governed by a $(G,L)$ continuum as $1/(|G|\cdot L)$. The processing gain $10\log_{10}(M)$ dB equals the classical beamforming gain, establishing that this gain is a property of group order, not sensor count. The DFT, DCT, and KLT are unified as group-matched special cases. We conjecture a General Algebraic Averaging Theorem extending these results to arbitrary statistics, with variance governed by the effective group order $d_{\mathrm{eff}}$. Monte Carlo experiments on the first four sample moments across five group types confirm the conjecture to four-digit precision. The framework exploits the $structure$ of information (representation-theoretic symmetry of the data object) rather than the content, complementing Shannon's theory. Five applications are demonstrated: single-snapshot MUSIC, massive MIMO, single-pulse waveform classification, graph signal processing, and analysis of transformer LLMs. Techniques for blind group matching are described.

cs.LG↗
arXiv2026

Quantum Algebraic Diversity: Single-Copy Density Matrix Estimation via Group-Structured Measurements

We extend the algebraic diversity (AD) framework from classical signal processing to quantum measurement theory. The Quantum Algebraic Diversity (QAD) Theorem establishes that a group-structured positive operator-valued measure (POVM) applied to a single copy of a quantum state produces a full-rank, group-averaged density matrix estimator whose eigenbasis and eigenvalue ordering track those of the true density matrix, with a bias toward the symmetrized state, analogous to the classical recovery of covariance eigenstructure from a single observation. We establish a Classical-Quantum Duality Map connecting classical covariance estimation to quantum state tomography, and an Optimality Inheritance Theorem showing that classical group optimality transfers to quantum settings via the Born map within the group-averaged family. SIC-POVMs are identified as AD with the Heisenberg-Weyl group and mutually unbiased bases as AD with the Clifford group, revealing the hierarchy $\mathrm{HW}(d) \subseteq \mathcal{C}(d) \subseteq S_d$ that mirrors the classical $\mathbb{Z}_M \subseteq G_{\min} \subseteq S_M$. The double-commutator eigenvalue theorem gives polynomial-time adaptive POVM selection. A worked qubit example shows the group-averaged estimator from a single computational-basis measurement, averaged over a matched $\mathbb{Z}_2$ group, reaching fidelity 0.99 where standard single-basis tomography gives a rank-1 estimate of fidelity 0.80. Monte Carlo simulations for $d = 2$ to $13$ confirm fidelity above 0.90 from a single outcome while standard fidelity degrades as $\sim 1/d$. The growing ratio reflects collapse of the rank-1 standard estimator, not fewer copies per parameter: the biased single-copy estimator reduces the number of distinct measurement settings, not the per-parameter sampling cost, and a genuine copy reduction holds only under exact symmetry.

quant-ph↗
arXiv2025

Parallel Architecture of a Frequency Comb Qudit Quantum Processor

Quantum optical frequency combs provide an intriguing approach to high-dimensional quantum states. Because of the need to move probabilities among different colors, the realization of gates appropriate to multicolor photons requires nonlinear or electro-optic mixing. This paper describes a novel architecture for such gates. The parallel arrangement of mixers by dimension allows graceful scaling beyond two-dimensional qubits. The parallelism of the implementation simplifies the programming of the gate for a particular operation. As an example, we demonstrate the design of a four-dimensional Chrestenson operator.

quant-ph↗
arXiv2025

Programming Quantum Computers with Large Language Models

Large language models (LLMs) promise transformative change to fields as diverse as medical diagnosis, legal services, and software development. One reason for such an impact is LLMs' ability to make highly technical endeavors more accessible to a broader audience. Accessibility has long been a goal for the growing fields of quantum computing, informatics, and engineering, especially as more quantum systems become publicly available via cloud interfaces. Between programming quantum computers and using LLMs, the latter seems the more accessible task: while leveraging an LLM's fullest potential requires experience with prompt engineering, any literate person can provide queries and read responses. By contrast, designing and executing quantum programs -- outside of those available online -- requires significant background knowledge, from selection of operations for algorithm implementation to configuration choices for particular hardware specifications and providers. Current research is exploring LLM utility for classical software development, but there has been relatively little investigation into the same for quantum programming. Consequently, this work is a first look at how well an uncustomized, publicly available LLM can write straightforward quantum circuits. We examine how well OpenAI's ChatGPT (GPT-4) can write quantum circuits for two hardware providers: the superconducting qubit machines of IBM and the photonic devices of Xanadu. We find that ChatGPT currently fares substantially better with the former.

quant-ph↗
arXiv2024

Automated Quantum Circuit Generation for Computing Inverse Hash Functions

Several cryptographic systems depend upon the computational difficulty of reversing cryptographic hash functions. Robust hash functions transform inputs to outputs in such a way that the inputs cannot be later retrieved in a reasonable amount of time even if the outputs and the function that created them are known. Consequently, hash functions can be cryptographically secure, and they are employed in encryption, authentication, and other security methods. It has been suggested that such cryptographically-secure hash functions will play a critical role in the era of post-quantum cryptography (PQC), as they do in conventional systems. In this work, we introduce a procedure that leverages the principle of reversibility to generate circuits that invert hash functions. We provide a proof-of-concept implementation and describe methods that allow for scaling the hash function inversion approach. Specifically, we implement one manifestation of the algorithm as part of a more general automated quantum circuit synthesis, compilation, and optimization toolkit. We illustrate production of reversible circuits for crypto-hash functions that inherently provide the inverse of the function, and we describe data structures that increase the scalability of the hash function inversion approach.

quant-ph↗
arXiv2024

Designing a Photonic Physically Unclonable Function Having Resilience to Machine Learning Attacks

Physically unclonable functions (PUFs) are designed to act as device 'fingerprints.' Given an input challenge, the PUF circuit should produce an unpredictable response for use in situations such as root-of-trust applications and other hardware-level cybersecurity applications. PUFs are typically subcircuits present within integrated circuits (ICs), and while conventional IC PUFs are well-understood, several implementations have proven vulnerable to malicious exploits, including those perpetrated by machine learning (ML)-based attacks. Such attacks can be difficult to prevent because they are often designed to work even when relatively few challenge-response pairs are known in advance. Hence the need for both more resilient PUF designs and analysis of ML-attack susceptibility. Previous work has developed a PUF for photonic integrated circuits (PICs). A PIC PUF not only produces unpredictable responses given manufacturing-introduced tolerances, but is also less prone to electromagnetic radiation eavesdropping attacks than a purely electronic IC PUF. In this work, we analyze the resilience of the proposed photonic PUF when subjected to ML-based attacks. Specifically, we describe a computational PUF model for producing the large datasets required for training ML attacks; we analyze the quality of the model; and we discuss the modeled PUF's susceptibility to ML-based attacks. We find that the modeled PUF generates distributions that resemble uniform white noise, explaining the exhibited resilience to neural-network-based attacks designed to exploit latent relationships between challenges and responses. Preliminary analysis suggests that the PUF exhibits similar resilience to generative adversarial networks, and continued development will show whether more-sophisticated ML approaches better compromise the PUF and -- if so -- how design modifications might improve resilience.

cs.CR↗
arXiv2024

A Photonic Physically Unclonable Function's Resilience to Multiple-Valued Machine Learning Attacks

Physically unclonable functions (PUFs) identify integrated circuits using nonlinearly-related challenge-response pairs (CRPs). Ideally, the relationship between challenges and corresponding responses is unpredictable, even if a subset of CRPs is known. Previous work developed a photonic PUF offering improved security compared to non-optical counterparts. Here, we investigate this PUF's susceptibility to Multiple-Valued-Logic-based machine learning attacks. We find that approximately 1,000 CRPs are necessary to train models that predict response bits better than random chance. Given the significant challenge of acquiring a vast number of CRPs from a photonic PUF, our results demonstrate photonic PUF resilience against such attacks.

cs.CR↗
arXiv2024

Demonstration of a Hardware-Independent Toolkit for Automated Quantum Subcircuit Synthesis

The quantum computer has become contemporary reality, with the first two-qubit machine of mere decades ago transforming into cloud-accessible devices with tens, hundreds, or -- in a few cases -- even thousands of qubits. While such hardware is noisy and still relatively small, the increasing number of operable qubits raises another challenge: how to develop the now-sizeable quantum circuits executable on these machines. Preparing circuits manually for specifications of any meaningful size is at best tedious and at worst impossible, creating a need for automation. This article describes an automated quantum-software toolkit for synthesis, compilation, and optimization, which transforms classically-specified, irreversible functions into both technology-independent and technology-dependent quantum circuits. We also describe and analyze the toolkit's application to three situations -- quantum read-only memories, quantum random number generators, and quantum oracles -- and illustrate the toolkit's start-to-finish features, from the input of classical functions to the output of technology-dependent quantum circuits. Furthermore, we illustrate how the toolkit enables research beyond circuit synthesis, including comparison of synthesis and optimization methods and deeper understanding of even well-studied quantum algorithms. As quantum hardware continues to develop, such quantum circuit toolkits will play a critical role in realizing its potential.

quant-ph↗