arXiv subjects

Steven T. Flammia

Publications and source records attributed to Steven T. Flammia.

At least 19 recordsLinked to original sources

arXiv2026

Autonomous Hamiltonian certification and changepoint detection

Modern quantum devices require high-precision Hamiltonian dynamics, but environmental noise can cause calibrated Hamiltonian parameters to drift over time, necessitating expensive recalibration. Detecting when recalibration is needed is challenging, especially since the very gates required for sophisticated verification protocols may themselves be miscalibrated. While cloud quantum computing services implement heuristic routines for triggering recalibration, the fundamental limits of optimal recalibration are not yet known. We develop efficient Hamiltonian certification and changepoint detection protocols in the autonomous setting, where we cannot rely on an external noiseless device and use only single-qubit gates and measurements, making the protocols robust to the calibration issues for multi-qubit operations they aim to detect. For unknown $n$-qubit Hamiltonians $H$ and $H_0$ with operator norm bounded by $M$, our certification protocol distinguishes whether $\|H-H_0\|_F\geqε$ or $\|H-H_0\|_F\leq O(ε/\sqrt{n})$ with sample complexity $O(nM^2\ln(1/δ)/ε^2)$ and total evolution time $O(nM\ln(1/δ)/ε^2)$. We achieve this by evolving random stabilizer product states and performing adaptive single-qubit measurements based on a classically simulable hypothesis state. Extending this to continuous monitoring, we develop an online changepoint detection algorithm using the CUSUM procedure that achieves a detection delay time bound of $O(nM\ln(M\mathbb{E}_\infty[T])/ε^2)$, matching the known asymptotically optimal scaling with respect to false alarm run time $\mathbb{E}_\infty[T]$. Our approach enables quantum devices to autonomously monitor their own calibration status without requiring ancillary systems, entangling operations, or a trusted reference device, offering a practical solution for robust quantum computing with contemporary noisy devices.

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arXiv2026

Fermionic Averaged Circuit Eigenvalue Sampling

Fermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.

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arXiv2026

Onset of Ergodicity Across Scales on a Digital Quantum Processor

Understanding how isolated quantum many-body systems thermalize remains a central question in modern physics. We study the onset of ergodicity in a two-dimensional disordered Heisenberg Floquet model using digital quantum simulation on IBM's Nighthawk superconducting processor, reaching system sizes of up to $10\times10$ qubits. We probe ergodicity across different length scales by coarse-graining the system into spatial patches of varying sizes and introducing a measure based on the collision entropy of each patch, enabling a detailed study of when ergodic behavior emerges across scales. The high sampling rate of superconducting quantum processing units, together with an optimal sample estimator, allow us to access patches of sizes up to $3\times3$. We observe that as the Heisenberg coupling $J$ increases, the noiseless system undergoes a smooth crossover from subergodic to ergodic behavior, with smaller patches approaching their random-matrix-theory values first, thereby revealing a hierarchy across scales. In the region of parameter space where classical tensor-network simulations are reliable, small patches or small values of $J$, we find excellent agreement with the error-mitigated quantum simulation. Beyond this regime, volume-law entanglement and contraction complexity growth causes the cost of classical methods to rise sharply. Our results open new directions for the use of quantum computers in the study of quantum thermalization.

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arXiv2026

Quantum Phaselift

Estimating quantum time-series such as the Loschmidt amplitude $f(t)=\langleψ|\mathrm{e}^{-\mathrm{i}Ht}|ψ\rangle$ is central to spectroscopy, Hamiltonian analysis, and many phase-estimation algorithms. Direct estimation via the Hadamard test requires controlled implementations of $\mathrm{e}^{-\mathrm{i}Ht}$, and the depth of these controlled circuits grows with $t$, making long-time estimation challenging on near-term hardware. We introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix $Z = f f^\dagger$ rather than estimating $f$ directly. We propose simple quantum circuits for estimating the entries of $Z$ and show that measuring only a narrow band of this matrix around the diagonal is sufficient to uniquely recover $f$. Crucially, this reformulation decouples the controlled circuit depth from the maximum evolution time to scale instead with the width of the measured band. We prove that a $O(1)$ bandwidth suffices for generic signals, leading to substantial savings in controlled operations compared to direct estimation methods. We develop three recovery algorithms with provable exact recovery in the noiseless setting and stability under measurement noise. Finally, we numerically demonstrate that high-quality recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model signals of size exceeding 100 time points using only a few million measurement shots and reasonable post-processing time, making our time-series estimation techniques efficient and effective for near-term implementations.

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arXiv2026

Chasing shadows with Gottesman-Kitaev-Preskill codes

We consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.

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arXiv2025

Fiber Bundle Fault Tolerance of GKP Codes

We investigate multi-mode GKP (Gottesman--Kitaev--Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman--Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.

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arXiv2025

Ansatz-free Hamiltonian learning with Heisenberg-limited scaling

Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term \emph{ansatz-free Hamiltonian learning}, remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system's real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.

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arXiv2025

Efficient self-consistent learning of gate set Pauli noise

Understanding quantum noise is an essential step towards building practical quantum information processing systems. Pauli noise is a useful model that has been widely applied in quantum benchmarking, error mitigation, and error correction. Despite intensive study, most existing works focus on learning Pauli noise channels associated with some specific gates rather than treating the gate set as a whole. A learning algorithm that is self-consistent, complete, and efficient at the same time is yet to be established. In this work, we study the task of gate set Pauli noise learning, where a set of quantum gates, state preparation, and measurements all suffer from unknown Pauli noise channels with a customized noise ansatz. Using tools from algebraic graph theory, we analytically characterize the self-consistently learnable degrees of freedom for Pauli noise models with arbitrary linear ansatz, and design experiments to efficiently learn all the learnable information. Specifically, we show that all learnable information about the gate noise can be learned to relative precision, under mild assumptions on the noise ansatz. We then demonstrate the flexibility of our theory by applying it to concrete physically motivated ansatzs (such as spatially local or quasi-local noise) and experimentally relevant gate sets (such as parallel CZ gates). These results not only enhance the theoretical understanding of quantum noise learning, but also provide a feasible recipe for characterizing existing and near-future quantum information processing devices.

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arXiv2024

Learning $k$-body Hamiltonians via compressed sensing

We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $\epsilon$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/\epsilon)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.

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arXiv2024

Lattices, Gates, and Curves: GKP codes as a Rosetta stone

Gottesman-Kitaev-Preskill (GKP) codes are a promising candidate for implementing fault tolerant quantum computation in quantum harmonic oscillator systems such as superconducting resonators, optical photons and trapped ions, and in recent years theoretical and experimental evidence for their utility has steadily grown. It is known that logical Clifford operations on GKP codes can be implemented fault tolerantly using only Gaussian operations, and several theoretical investigations have illuminated their general structure. In this work, we explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show how they are identified with the mapping class group of suitable genus $n$ surfaces. This correspondence introduces a topological interpretation of fault tolerance for GKP codes and motivates the connection between GKP codes (lattices), their Clifford gates, and algebraic curves, which we explore in depth. For a single-mode GKP code, we identify the space of all GKP codes with the moduli space of elliptic curves, given by the three sphere with a trefoil knot removed, and explain how logical degrees of freedom arise from the choice of a level structure on the corresponding curves. We discuss how the implementation of Clifford gates corresponds to homotopically nontrivial loops on the space of all GKP codes and show that the modular Rademacher function describes a topological invariant for certain Clifford gates implemented by such loops. Finally, we construct a universal family of GKP codes and show how it gives rise to an explicit construction of fiber bundle fault tolerance as proposed by Gottesman and Zhang for the GKP code. On our path towards understanding this correspondence, we introduce a general algebraic geometric perspective on GKP codes and their moduli spaces, which uncovers a map towards many possible routes of future research.

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arXiv2024

Quantum chi-squared tomography and mutual information testing

For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\widetilde{O}(r^{.5}d^{1.5}/ε) \leq \widetilde{O}(d^2/ε)$ copies suffice for accuracy~$ε$ with respect to (Bures) $χ^2$-divergence, and $\widetilde{O}(rd/ε)$ copies suffice for accuracy~$ε$ with respect to quantum relative entropy. The best previous bound was $\widetilde{O}(rd/ε) \leq \widetilde{O}(d^2/ε)$ with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the $χ^2$-divergence. For algorithms that are required to use single-copy measurements, we show that $\widetilde{O}(r^{1.5} d^{1.5}/ε) \leq \widetilde{O}(d^3/ε)$ copies suffice for $χ^2$-divergence, and $\widetilde{O}(r^{2} d/ε)$ suffice for relative entropy. Using this tomography algorithm, we show that $\widetilde{O}(d^{2.5}/ε)$ copies of a $d\times d$-dimensional bipartite state suffice to test if it has quantum mutual information~$0$ or at least~$ε$. As a corollary, we also improve the best known sample complexity for the \emph{classical} version of mutual information testing to $\widetilde{O}(d/ε)$.

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arXiv2024

Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons

Quantum error correction with erasure qubits promises significant advantages over standard error correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing the qubit. We demonstrate that a "dual-rail qubit" consisting of a pair of resonantly coupled transmons can form a highly coherent erasure qubit, where transmon $T_1$ errors are converted into erasure errors and residual dephasing is strongly suppressed, leading to millisecond-scale coherence within the qubit subspace. We show that single-qubit gates are limited primarily by erasure errors, with erasure probability $p_\text{erasure} = 2.19(2)\times 10^{-3}$ per gate while the residual errors are $\sim 40$ times lower. We further demonstrate mid-circuit detection of erasure errors while introducing $< 0.1\%$ dephasing error per check. Finally, we show that the suppression of transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range, offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail qubits as an attractive building block for hardware-efficient quantum error correction.

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arXiv2024

Clifford-deformed Surface Codes

Various realizations of Kitaev's surface code perform surprisingly well for biased Pauli noise. Attracted by these potential gains, we study the performance of Clifford-deformed surface codes (CDSCs) obtained from the surface code by applying single-qubit Clifford operators. We first analyze CDSCs on the $3\times 3$ square lattice and find that, depending on the noise bias, their logical error rates can differ by orders of magnitude. To explain the observed behavior, we introduce the effective distance $d'$, which reduces to the standard distance for unbiased noise. To study CDSC performance in the thermodynamic limit, we focus on random CDSCs. Using the statistical mechanical mapping for quantum codes, we uncover a phase diagram that describes random CDSC families with $50\%$ threshold at infinite bias. In the high-threshold region, we further demonstrate that typical code realizations outperform the thresholds and subthreshold logical error rates, at finite bias, of the best-known translationally invariant codes. We demonstrate the practical relevance of these random CDSC families by constructing a translation-invariant CDSC belonging to a high-performance random CDSC family. We also show that our translation-invariant CDSC outperforms well-known translation-invariant CDSCs such as the XZZX and XY codes.

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arXiv2023

Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes

Low-depth random circuit codes possess many desirable properties for quantum error correction but have so far only been analyzed in the code capacity setting where it is assumed that encoding gates and syndrome measurements are noiseless. In this work, we design a fault-tolerant distillation protocol for preparing encoded states of one-dimensional random circuit codes even when all gates and measurements are subject to noise. This is sufficient for fault-tolerant quantum memory since these encoded states can then be used as ancillas for Steane error correction. We show through numerical simulations that our protocol can correct erasure errors up to an error rate of $2\%$. In addition, we also extend results in the code capacity setting by developing a maximum likelihood decoder for depolarizing noise similar to work by Darmawan et al. As in their work, we formulate the decoding problem as a tensor network contraction and show how to contract the network efficiently by exploiting the low-depth structure. Replacing the tensor network with a so-called ''tropical'' tensor network, we also show how to perform minimum weight decoding. With these decoders, we are able to numerically estimate the depolarizing error threshold of finite-rate random circuit codes and show that this threshold closely matches the hashing bound even when the decoding is sub-optimal.

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arXiv2023

Improved decoding of circuit noise and fragile boundaries of tailored surface codes

Realizing the full potential of quantum computation requires quantum error correction (QEC), with most recent breakthrough demonstrations of QEC using the surface code. QEC codes use multiple noisy physical qubits to encode information in fewer logical qubits, enabling the identification of errors through a decoding process. This process increases the logical fidelity (or accuracy) making the computation more reliable. However, most fast (efficient runtime) decoders neglect important noise characteristics, thereby reducing their accuracy. In this work, we introduce decoders that are both fast and accurate, and can be used with a wide class of QEC codes including the surface code. Our decoders, named belief-matching and belief-find, exploit all noise information and thereby unlock higher accuracy demonstrations of QEC. Using the surface code threshold as a performance metric, we observe a threshold at 0.94\% error probability for our decoders, outperforming the 0.82\% threshold for a standard minimum-weight perfect matching decoder. We also tested our belief-matching decoders in a theoretical case study of codes tailored to a biased noise model. We find that the decoders led to a much higher threshold and lower qubit overhead in the tailored surface code with respect to the standard, square surface code. Surprisingly, in the well-below threshold regime, the rectangular surface code becomes more resource-efficient than the tailored surface code, due to a previously unnoticed phenomenon that we call "fragile boundaries". Our decoders outperform all other fast decoders in terms of threshold and accuracy, enabling better results in current quantum error correction experiments and opening up new areas for theoretical case studies.

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arXiv2023

Learning correlated noise in a 39-qubit quantum processor

Building error-corrected quantum computers relies crucially on measuring and modeling noise on candidate devices. In particular, optimal error correction requires knowing the noise that occurs in the device as it executes the circuits required for error correction. As devices increase in size we will become more reliant on efficient models of this noise. However, such models must still retain the information required to optimize the algorithms used for error correction. Here we propose a method of extracting detailed information of the noise in a device running syndrome extraction circuits. We introduce and execute an experiment on a superconducting device using 39 of its qubits in a surface code doing repeated rounds of syndrome extraction, but omitting the mid-circuit measurement and reset. We show how to extract from the 20 data qubits the information needed to build noise models of various sophistication in the form of graphical models. These models give efficient descriptions of noise in large-scale devices and are designed to illuminate the effectiveness of error correction against correlated noise. Our estimates are furthermore precise: we learn a consistent global distribution where all one- and two-qubit error rates are known to a relative error of 0.1%. By extrapolating our experimentally learned noise models towards lower error rates, we demonstrate that accurate correlated noise models are increasingly important for successfully predicting sub-threshold behavior in quantum error correction experiments.

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arXiv2022

Foundations for learning from noisy quantum experiments

Understanding what can be learned from experiments is central to scientific progress. In this work, we use a learning-theoretic perspective to study the task of learning physical operations in a quantum machine when all operations (state preparation, dynamics, and measurement) are a priori unknown. We prove that, without any prior knowledge, if one can explore the full quantum state space by composing the operations, then every operation can be learned. When one cannot explore the full state space but all operations are approximately known and noise in Clifford gates is gate-independent, we find an efficient algorithm for learning all operations up to a single unlearnable parameter characterizing the fidelity of the initial state. For learning a noise channel on Clifford gates to a fixed accuracy, our algorithm uses quadratically fewer experiments than previously known protocols. Under more general conditions, the true description of the noise can be unlearnable; for example, we prove that no benchmarking protocol can learn gate-dependent Pauli noise on Clifford+T gates even under perfect state preparation and measurement. Despite not being able to learn the noise, we show that a noisy quantum computer that performs entangled measurements on multiple copies of an unknown state can yield a large advantage in learning properties of the state compared to a noiseless device that measures individual copies and then processes the measurement data using a classical computer. Concretely, we prove that noisy quantum computers with two-qubit gate error rate $ε$ can achieve a learning task using $N$ copies of the state, while $N^{Ω(1/ε)}$ copies are required classically.

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arXiv2022

Tailored XZZX codes for biased noise

Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.

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