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Gate Set Tomography

Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, Robin Blume-Kohout

arXiv:2009.07301v2quant-ph

TL;DR

Quantum processors need detailed characterization of individual logic operations for debugging, device modeling, and predicting complex circuit performance. This paper develops GST's foundations, including direct gate-set fitting and long-sequence protocols, and describes its comprehensive characterization, precision, and analytical limitations.

  • Problem

    Detailed tomography of individual quantum gates remains necessary for debugging, rigorous device modeling, and reliable prediction of complex circuit performance.

  • Method

    GST fits gate-set models directly to observed quantum-circuit outcome frequencies and uses long-sequence protocols for efficient estimation.

  • Results

    GST is presented as a powerful and comprehensive technique for detailed characterization of individual logic operations, with deep periodic circuits enabling Heisenberg-like precision scaling modulo gauge freedom.

  • Takeaways & Limitations

    GST supports debugging, rigorous device modeling, and reliable prediction of complex circuit performance, including quantum-error-correction circuits.

  • Takeaways & Limitations

    The LAN-based ellipsoid confidence regions can overestimate uncertainty and occasionally provide insufficient coverage when LAN is a poor approximation.

Abstract

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Gate set tomography (GST) is a protocol for detailed, predictive characterization of logic operations (gates) on quantum computing processors. Early versions of GST emerged around 2012-13, and since then it has been refined, demonstrated, and used in a large number of experiments. This paper presents the foundations of GST in comprehensive detail. The most important feature of GST, compared to older state and process tomography protocols, is that it is calibration-free. GST does not rely on pre-calibrated state preparations and measurements. Instead, it characterizes all the operations in a gate set simultaneously and self-consistently, relative to each other. Long sequence GST can estimate gates with very high precision and efficiency, achieving Heisenberg scaling in regimes of practical interest. In this paper, we cover GST's intellectual history, the techniques and experiments used to achieve its intended purpose, data analysis, gauge freedom and fixing, error bars, and the interpretation of gauge-fixed estimates of gate sets. Our focus is fundamental mathematical aspects of GST, rather than implementation details, but we touch on some of the foundational algorithmic tricks used in the pyGSTi implementation.

1 Introduction

Quantum device characterization spans holistic benchmarking and detailed tomography, but rigorous modeling and prediction of individual logic operations require gate set tomography. This paper addresses the missing comprehensive theoretical account of GST, emphasizing its calibration-free, self-consistent characterization of complete gate sets.

  • Motivation: Fault-tolerant quantum computation requires high-fidelity operations whose errors satisfy conditions assumed by threshold theorems.Failure rates as low as 10^-4 are described as rare and remarkable, motivating precise characterization.
  • Motivation: QCVV protocols run quantum-circuit experiments, process their data, and estimate aspects of processor behavior with varying levels of detail.Some methods provide predictive models of logic operations, whereas others provide semipredictive partial characterizations or holistic summaries.
  • Gate set tomography: GST is currently the most powerful and comprehensive technique for detailed tomography of individual logic operations, and this paper fills the literature’s gap in comprehensive GST theory.The paper traces GST’s development since 2012–13, subsequent experiments, and implementation in open-source software.
  • Gate set tomography: GST reconstructs an entire gate set almost entirely without calibration, avoiding dependence on prior descriptions of prepared states or measurements.Its calibration-free characterization is linked to estimating state preparations, measurements, and logic gates simultaneously and self-consistently.
  • Calibration: Standard state and process tomography can become systematically unreliable when their pre-calibrated reference-frame operations are unknown or misidentified.Errors in state preparation and measurement can dominate process-tomography inaccuracy in realistic scenarios.
  • Scope: The paper is a theory-focused treatment of GST’s mathematical background, justifications, and derivations rather than a practical how-to manual.It covers foundations, LGST, long-sequence GST, likelihood maximization, gauge freedom, and data interpretation, with implementation discussed only incidentally.

2 Background

The paper models fixed-input, classical-output quantum circuits as initialization, circuit-layer gates, and measurement, using Hilbert–Schmidt-space representations for states, effects, and operations. It then motivates tomography and highlights practical limitations from finite data and imperfect calibration.

  • Quantum processors and circuits: Fixed-input, classical-output circuits initialize all qubits, apply zero or more gates, and finish by measuring all qubits.The paper treats each circuit layer as an independent n-qubit gate, rather than modeling parallel elementary operations separately.
  • Quantum processors and circuits: Simultaneous one- and two-qubit elementary operations are grouped into circuit layers that act on all qubits.This layer-based description becomes difficult to scale because exponentially many distinct layers can be possible as the qubit count grows.
  • Mathematical representation: The paper’s QCVV models include adjustable representations of initialization, gates, and measurements that predict circuit outcome probabilities.GST uses Hilbert–Schmidt space to represent these operations and the gate set collectively.
  • Mathematical representation: For n qubits, the Hilbert-space dimension is d = 2^n, and tensor products of Pauli operators provide a suitable basis for Hilbert–Schmidt representations.The paper assumes finite-dimensional Hilbert spaces and uses a Hermitian basis for the real subspace containing Hermitian operators.
  • Mathematical representation: Quantum states are positive semidefinite, trace-1 density matrices, while measurement outcomes are represented by POVM effects satisfying positivity and normalization conditions.States and effects are represented in Hilbert–Schmidt space using superket and superbra notation.
  • State tomography: Finite samples make observed frequencies differ from the corresponding probabilities, so probability estimates are required in practical state tomography.Linear inversion is a theoretical cornerstone but can be inaccurate and can produce estimates that are not positive.
  • Tomography: Standard process tomography relies on known informationally complete fiducial states and state tomography of outputs, while measurement tomography relies on pre-calibrated fiducial states.In realistic scenarios, state-preparation and measurement errors can dominate process-tomography inaccuracy.

3 Linear gate set tomography (LGST)

LGST provides a calibration-free, linear-algebra-based estimate of an entire gate set, reconstructing all gates self-consistently up to a shared gauge transformation. It is useful as a fast initialization for higher-accuracy maximum-likelihood GST, while standalone LGST has process-tomography-like scaling and can be impractical at high precision.

  • Purpose and calibration: LGST combines state, process, and measurement tomography without pre-calibrated reference frames, producing a low-precision estimate of the whole gate set.Its self-calibrating design avoids the need to know effective preparations and measurements in advance.
  • Experiment design: LGST prescribes a structured set of short circuits whose probabilities can be analyzed using linear algebra rather than an unstructured nonlinear fit.This design avoids the difficult optimization landscape encountered by early maximum-likelihood approaches using short circuits alone.
  • Accuracy and cost: N ≈ 10^10 repetitions per circuit are required to estimate a gate set within ±10^-5 using LGST, making standalone use impractical.LGST estimation error scales as O(1/sqrt(N)), the same scaling as process tomography.
  • Gauge freedom: LGST estimates every gate using the same unknown similarity transformation, thereby recovering the entire gate set up to a global gauge.Gauge-equivalent gate sets produce identical observable probabilities, so the transformation is unknowable but irrelevant to prediction.
  • Maximum likelihood refinement: The paper reframes gate-set reconstruction as fitting a parameterized gate-set model to observed circuit data, with least squares treated as a convenient but imperfect approximation.Maximum likelihood is presented as more accurate when the likelihood is nonlinear.
  • Maximum likelihood refinement: LGST remains important because it supplies a fast, reasonably good starting estimate for numerical maximum-likelihood fitting.The likelihood is nonlinear and can contain local extrema, so optimization benefits from LGST initialization.

4 Long-sequence Gate Set Tomography

Long-sequence GST improves gate-set estimation by designing deep, structured circuits that amplify gate deviations while retaining fiducial-sandwich tomography. The resulting experiment supports accuracy scaling inversely with circuit depth, although likelihood optimization remains highly non-convex.

  • Motivation: Long-sequence GST uses deep circuits whose outcome probabilities become more sensitive to repeatedly applied gate elements, enabling higher estimation precision.Sensitivity can grow with the number of gate appearances in a circuit.
  • Historical evolution: Random circuits achieved precision that declined as O(1/… with increasing depth, motivating a more structured experiment design.
  • Historical evolution: eLGST replaced random circuits with base circuits formed from repeated germs, whose collectively amplified deviations support consistent accuracy scaling as 1/L.Germs are selected to amplify different deviations so that all deviations are amplified collectively.
  • Experiment design: GST experiment designs use germs, near-logarithmically spaced powers, and fiducial pairs to probe amplified parameters across circuit depths.
  • Experiment design: Each GST circuit sandwiches a germ-derived base circuit between preparation and measurement fiducials, preserving LGST’s basic structure while amplifying errors.
  • Parameter estimation: The GST likelihood is extremely non-convex because repeated gates make outcome probabilities nonlinear and oscillatory, producing many local maxima.

5 Advanced long-sequence GST

Advanced long-sequence GST treats gate sets as parameterized statistical models and uses constraints, gauge analysis, and fiducial-pair reduction to shape estimation. These tools can enforce physical structure and reduce experiment size, but complete positivity interacts nontrivially with gauge freedom and reduction lacks systematic robustness validation.

  • Gate set models: A gate set model maps a parameter space to gate sets, allowing long-sequence GST to optimize over constrained subsets of matrix space.Different models can restrict the optimization to physically motivated families such as CPTP gate sets.
  • Constrained models: Trace preservation is straightforward to impose, and the TP parameterized model generally outperforms the fully parameterized model without adding complications.Its gauge transformations are exactly the TP transformations, leaving d2 − d gauge parameters.
  • Constrained models: Complete positivity is harder to impose because it is a nonlinear inequality rather than a constant equality.Cholesky-based Choi representations and Lindbladian error generators are described as ways to construct CP-constrained models.
  • Gauge freedom: Gauge transformations preserve observable probabilities but non-unitary transformations can change complete positivity, so CP status is gauge-dependent.
  • Fiducial-pair reduction: Fiducial-pair reduction can shrink experiments by 50-90% in many cases, but its robustness remains unstudied and reduction can increase local-maximum risk.

6 Analyzing GST estimates

GST analysis first evaluates whether the estimated gate set fits the observed data, then interprets the estimate through gauge-aware metrics and diagnostics.

  • Goodness-of-fit: GST goodness-of-fit compares the estimate’s likelihood with a maximal model to test whether the data are consistent with a Markovian gate-set model.Wilks’ theorem provides the likelihood-ratio accounting for the models’ differing numbers of free parameters.
  • Goodness-of-fit: Nσ ≫1 indicates significant model violation and evidence that no gate set can describe all the data under the assumed model.Such violation is interpreted as evidence of non-Markovian noise, including effects such as drift, leakage, persistent environments, spillover, or heating.
  • Goodness-of-fit: A large Nσ establishes high confidence that the Markovian model was violated, but does not quantify the required model expansion or make the GST estimate useless.If any model violation exists, Nσ generally increases linearly with the number of circuit repetitions.
  • Goodness-of-fit: Per-circuit likelihood contributions identify which circuits are inconsistent with the GST estimate, but multiple simultaneous tests require an adjusted detection threshold.For K parallel tests, each test should use the α1/K confidence level to keep the probability of any false positive below α.
  • Goodness-of-fit: Figure 6 organizes per-circuit goodness-of-fit values by germ and base-circuit depth, helping reveal patterns of model violation across circuits.Deeper circuits are more sensitive to most forms of non-Markovian noise, and red boxes clustered by germ can indicate germ-specific effects.
  • Gauge optimization: Gauge transformations preserve circuit outcome probabilities but can substantially change individual gate matrices and derived metrics such as fidelity.Because common metrics are often gauge-variant, GST uses gauge optimization to select a representation close to ideal target gates.

2. Trace/diamond distance,

Gauge optimization is necessary because gate-set estimates are gauge-equivalent rather than uniquely represented, and metric choice affects whether highly precise gate estimates are misleadingly polluted by SPAM uncertainty.

  • Metric choice: Infidelity is unsuitable for unconstrained gauge optimization because it may exceed one and need not be uniquely minimized at the target gate set.The problem is avoided only when complete positivity is imposed, which creates practical complications.
  • Metric choice: Squared Frobenius distance is preferred for gauge optimization because it is fast to compute, well-behaved, and usually yields few disadvantages.Trace/diamond distance is operationally meaningful but expensive, nonsmooth, and not significantly different in results.
  • Weighting: Equal weighting can assign about 0.5 × 10^-2 coherent error to both gates and SPAM, despite expected gate errors of ±10^-4 and SPAM errors of ±10^-2.This gauge choice is misleading because it transfers SPAM estimation noise into the more precise gate estimates.
  • Weighting: Weights should reflect operation frequency: the rule of thumb assigns α_i = γ_m,i = 1 for SPAM and β_i = 10^4 for gates when gates appear up to L = 100 times.Sequential optimization stages provide another heuristic, including restricting later optimization to unitary transformations acting on gates.
  • Gauge optimization: Gauge optimization converts an estimate into a fixed, effectively standard form so gauge-variant metrics such as fidelity or diamond distance can be interpreted.The optimized estimate remains gauge-equivalent to the original gate set.
  • Error bars: GST error bars can be obtained by bootstrapping or likelihood-ratio confidence regions, but the authors describe current algorithms as insufficiently developed.Parametric bootstrap is theoretically expected to be more reliable because it is less affected by constraint-induced bias and small-sample effects.

7 Summary

GST characterizes a quantum logic device as a self-consistent gate set rather than as unrelated independently estimated components. Its distinguishing features include calibration-free operation, high precision from deep circuits, flexible constraints, and built-in model validation.

  • GST is a new type of tomography that probes a gate set as an entity with properties distinct from unrelated component collections.
  • GST is calibration-free: it requires neither a pre-calibrated reference frame nor prior assumptions about operation accuracy.
  • GST produces self-consistent descriptions of state preparations, gates, and measurements relative to one another, exposing relational gauge-dependent errors.
  • Deep, periodic circuits give GST estimates Heisenberg-like precision scaling, modulo gauge freedom.
  • GST can incorporate physics-inspired constraints such as trace preservation and complete positivity while treating gate sets as parameterized models.
  • Overcomplete GST experiments support statistical detection and quantification of model violations as warnings for systematic characterization errors.

A.1 Gauge degrees of freedom

Gauge freedom creates families of gate-set parameters that produce identical observable probabilities, so data determine a best-fit equivalence class rather than a unique parameter point. These gauge fibers form a fiber-bundle structure, with model-dependent transformations and physically meaningful non-gauge directions.

  • Gauge transformations map gate sets to physically equivalent descriptions without changing observable probabilities or goodness of fit.
  • Gauge transformations form GL(d^2) for the fully parameterized model, a TP subgroup for TP models, and lack a well-defined global group under CP constraints.
  • For a single-qubit fully parameterized model there are 16 gauge directions, whereas a TP-parameterized model has 12.
  • The parameter space is foliated into gauge manifolds, with fibers representing equivalent gate sets and a base manifold representing non-gauge directions.
  • Infinitesimal gauge directions can be constructed from derivatives of matrix-space transformations and pulled back through the gate-set model mapping.
  • Data determine a best-fit fiber, while only non-gauge parameter combinations are physically meaningful and can have finite error bars.

A.2 Gauge considerations in error bars

Gauge directions make the likelihood Hessian rank-deficient, so uncertainty analysis must remove them before computing error bars. The selected non-gauge space or metric is not unique, and that choice affects the resulting error bars.

  • At the maximum-likelihood estimate, gauge directions have zero log-likelihood curvature, producing zero Hessian eigenvalues and preventing direct inversion.
  • The non-gauge directions are metric-dependent, so different valid metrics couple gauge and non-gauge motion differently and yield different error bars.
  • A metric can be optimized by minimizing a weighted sum of error-bar sizes for gate matrices and state-preparation or measurement-effect vectors.
  • Hessian-based error bars require regional gauge fixing around the maximum-likelihood orbit to project out gauge directions.
  • An alternative metric weights gate and SPAM parameters according to their intrinsic error rates.

B The extended LGST (eLGST) protocol

eLGST extends LGST with repeated germ circuits and iterative least-squares fitting to obtain high-precision gate estimates without unstable direct matrix roots. Its design addresses branch ambiguity and matrix-root instability while exploiting overcomplete fiducials.

  • eLGST applies LGST to repeated germ circuits at sequence lengths 1, 2, 4, 8, and beyond, then iteratively refines estimates using least-squares objectives.
  • eLGST avoids explicit high-order matrix roots and extracts high-precision gate estimates from refined germ estimates.
  • The algorithm was reliable and accurate on simulated data, motivating the stable long-sequence GST algorithm.
  • Iteratively increasing sequence length resolves scalar root branches with only logarithmic additional cost when enough data are available.
  • Direct matrix roots are unstable because matrix square-root functions can be nonsmooth and produce roots far from the true germ.
  • Overcomplete fiducials provide redundancy, precision, feasibility, and self-consistency checks, despite making the linear algebra non-square.
  • With finite-sample perturbations, eLGST seeks approximate gate solutions by minimizing least-squares residuals such as |Pk − AGkB|^2 and |P̃1l − AB|^2.

D Implementations in pyGSTi

The pyGSTi implementation uses algorithmic procedures to select informationally complete fiducials and prepare gate-set models for GST circuit construction. These implementation choices are tractable in practice while preserving the protocol’s required asymptotic accuracy scaling.

  • Implementation workflow: pyGSTi treats fiducial, germ, and germ-power selection as separate tasks, followed optionally by fiducial pair reduction.This decomposition is used in the most tested implementation for simplicity.
  • Gate-set model: The implementation uses a gate set model G whose parameters and circuit-selection properties should approximate the true or best-fit gate set.The model is initially set to a trace-preserving parameterization.
  • Fiducial selection: Fiducial selection chooses informationally complete effective state preparations and measurements represented by {Hk} and {Fk}.Informational completeness requires spanning B(H).
  • Fiducial selection: Candidate fiducial sets are evaluated through Gram-matrix singular values, maximizing the smallest of the top d^2 values to improve uniform linear independence.Sets with a small d^2-th singular value are close to linearly dependent and should be avoided.
  • Fiducial selection: Fiducial selection requires reasonably accurate gate estimates; discrepancies from the assumed gate set can be detected through empirical Gram-matrix singular values and remedied iteratively.The procedure relies on the actual gates being sufficiently close to those used during selection.

D.1.2 Germ Selection

Germ selection identifies short circuits whose repeated powers amplify gate-set parameter deviations, enabling long-sequence GST to estimate parameters with accuracy improving as circuit depth increases. The selection uses sensitivity analysis to cover all physically accessible nongauge directions while balancing amplification and noise.

  • Germ selection: Germ selection seeks low-depth circuits whose powers amplify useful operations of interest for tomography.The goal is to construct repeated circuits that expose gate-set parameters through their effect on measured probabilities.
  • Amplification and scaling: Increasing the maximum repetition depth l reduces parameter uncertainty as 1/l, yielding Heisenberg-like scaling that is better than increasing samples for a single gate.The total circuit count grows at worst linearly with l and often logarithmically because l-values are logarithmically spaced.
  • Error coverage: Single-gate powers do not amplify every error type: a tilt error can cancel after four repetitions, so additional germs such as GxGy are required.The composite germ GxGy amplifies the tilt deviation by p and estimates it with 1/(p√N) scaling.
  • Amplification and scaling: Repeating a germ power p amplifies suitable parameter deviations by p, reducing estimation uncertainty to 1/(p√N) in the phase-estimation example.Logarithmically spaced powers resolve branch ambiguities while retaining the improved scaling.
  • Completeness criterion: Candidate germs are ranked by Jacobian singular directions, whose singular values quantify amplification and whose zero values identify unamplified parameter combinations.A complete germ set has Jacobian rank equal to the number of physically accessible nongauge parameters.
  • Practical selection: In practice, germ selection assumes reversible gates and uses unitary projections or random unitary variations, while limiting useful depths according to stochastic or depolarizing noise.The score function estimates mean-squared error when a fixed number of counts is distributed across germs.

D.1.3 Base circuit selection

Base-circuit selection chooses germ powers and maximum depths, then combines them with fiducial circuits to form GST experiments. Fiducial pair reduction removes redundant circuits while targeting the same parameter amplification, although some proposed depth variations may reduce robustness.

  • Base-circuit construction: Long-sequence GST repeats each germ at logarithmically increasing depths until the repeated circuit reaches the noise-limited maximum depth Lη.The standard construction uses maximum depths and germ powers defined so repeated circuits do not exceed the chosen depth limits.
  • Base-circuit construction: A complete GST experiment combines fiducial circuits, germs, germ powers, and their associated outcome probabilities into circuits built from native gate-set operations.The construction tracks fiducials, germs, and germ powers separately for later data analysis.
  • Fiducial pair reduction: The standard circuit set contains numerical redundancy that stabilizes the optimization used for GST parameter estimation.Thus, reduction trades some redundancy for fewer prescribed circuits.
  • Fiducial pair reduction: Fiducial pair reduction exploits redundancy because each repeated germ needs only the amplified parameter combinations, not full tomographic information.These combinations are typically about d^2 rather than about d^3 circuits.
  • Fiducial pair reduction: Reducing the circuit count by roughly d^3/d^2 = d is expected to preserve the desired Heisenberg scaling.Per-germ fiducial pairs are selected so their derivative matrix reaches the maximal rank of parameters amplified by that germ.
  • Scope and caveat: Using different maximum-depth sets for different germs may preserve scaling on well-fit data but could reduce robustness on poorly fit data, a hypothesis left unverified by simulation.The authors identify this as future work rather than an established result.
  • Likelihood formulation: For non-trace-preserving gate sets, the likelihood uses independent Poisson outcome rates because predicted total counts need not equal observed counts.The resulting formulation also contains the trace-preserving likelihood as a special case up to an additive constant.

E Numerical verification

Numerical simulations verify the predicted accuracy scalings of LGST and long-sequence GST across perturbed one- and two-qubit gate sets. LGST follows sample-count scaling, while long-sequence GST follows depth scaling and translates to inverse-total-sample scaling.

  • LGST scaling: LGST reconstruction accuracy scales as O(1/√N), where N is the number of samples per circuit.The verification uses average diamond-norm distance after gauge optimization and finds slopes near -0.5 across tested one- and two-qubit gate sets.
  • Long-sequence GST scaling: Long-sequence GST reconstructs gate sets with O(1/L) accuracy, where L is the maximum depth of the repeated germ circuits.Figure 10 reports slopes close to -1 for one- and two-qubit simulations.
  • Verification procedure: The simulations generate perturbed imperfect gate sets, sample multinomial circuit outcomes, fit LGST or GST estimates, and compare log-log accuracy slopes with targets −1/2 and −1.The trials include 1-qubit and 2-qubit systems with random depolarization, SPAM error, and random rotations.
  • LGST scaling: Because LGST uses a fixed number of circuits, its accuracy also scales as O(1/√N′) with total sample count N′.Here N is proportional to N′.
  • Verification caveat: At large L, accuracy curves show an expected roll-off attributed to approaching Lη, where decoherence makes circuit outcomes uninformative.The bend is more visible in the two-qubit case.
  • Long-sequence GST scaling: Long-sequence GST’s O(1/L) scaling translates to O(1/N′) scaling when N is fixed and total samples grow proportionally with L.The proportionality argument pessimistically counts L distinct maximum-depth values, although the implementation usually uses log(L).
  • Verification setup: The tested implementation used LGST and long-sequence GST from pyGSTi version 0.9.9.2.Accuracy was quantified using gauge-optimized average diamond distance, though other well-behaved distances were expected to show the same scaling.

F χ2 estimator SPAM bias

Minimum-χ2 estimation can substantially overestimate rare-event probabilities and SPAM error in GST, while full MLE avoids these issues. The SPAM bias can indirectly underestimate the RB error rate.

  • 1.18×: When the true probability is 0.01, the average minimum-χ2 estimate is 0.0118.The estimator overestimates rare-event probabilities, with bias worsening as the true probability decreases.
  • 1.411p: For sufficiently small probabilities, the average minimum-χ2 estimate approaches approximately 1.411 times the true probability.
  • Many GST circuits with N ≈100 counts are combined into a single χ2 or likelihood function to estimate SPAM error.This makes the coin-bias example relevant to GST’s SPAM-error estimation procedure.
  • 100%: Minimum-χ2 estimation can overestimate SPAM error by as much as 100%.The bias typically does not directly affect gate estimates but can indirectly cause an underestimate of the RB error rate.
  • Full MLE generally provides accurate estimates and resolves the minimum-χ2 bias; its residual likelihood is χ2 distributed for valid models.The residual likelihood ratio also serves as an effective model-validation statistic.
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