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Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit
Robin Blume-Kohout, John King Gamble, Erik Nielsen, Jonathan Mizrahi, Jonathan D. Sterk, Peter Maunz
TL;DR
Standard tomography depends on calibrated preparations and measurements, though these may be implemented by the same gates under study, and conventional fidelity measures are problematic for unknown gate sets with gauge freedom. The paper introduces self-consistent gate set tomography, a closed-form linear-inversion starting point, and scoring based on predictions of held-out experiments. In a trapped-ion qubit demonstration, the GST estimates predicted test data well, while maximum-likelihood estimates were more accurate than LGST on short sequences.
Problem
Standard tomography assumes known preparations and measurements that may be implemented by the gates being characterized, while fidelity scoring lacks a true reference and is complicated by GST gauge freedom.
Source: B. Standard tomography · p. 2, §VI · p. 9Method
GST estimates a gate set self-consistently, using LGST as a closed-form first stage before maximum-likelihood refinement and scoring estimates by predictions on testing experiments.
Source: §IV · p. 7, §VII · p. 11Results
GST estimates were good predictors of test data, and maximum-likelihood methods were more accurate than LGST even for sequences of length 4 or 5.
Source: §1 · p. 11, §VII · p. 12Takeaways & Limitations
The scoring protocol evaluates a tomographic estimate by how well it predicts future data rather than by agreement with a possibly incorrect target or a potentially similarly biased estimate.
Source: §VII · p. 11The study does not impose complete positivity on estimated gates, instead allowing arbitrary matrices and relying on data consistency to ensure positive predictions.
Source: C. Positivity · p. 4
Abstract
from arXiv · showhide
We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characterize a set of Clifford-generating gates on a single trapped-ion qubit, and compare the performance of (i) standard process tomography; (ii) linear gate set tomography; and (iii) maximum likelihood gate set tomography.
I. BACKGROUND
Standard tomography represents states and processes through linear algebra, but it relies on known preparations and measurements to establish an absolute reference frame. In most quantum technologies, those preparations and measurements are themselves produced by imperfectly known gates.
A. Mathematical conventions
- Quantum processes act as linear maps on density matrices and can be represented as d^2 × d^2 superoperators in Hilbert-Schmidt space.States are positive semidefinite, unit-trace density matrices; measurements are POVMs whose effects sum to identity.Source: A. Mathematical conventions · p. 2
B. Standard tomography
- State tomography uses informationally complete measurements to estimate an unknown state, while process tomography also requires spanning test states and applies the unknown process to them.Both procedures estimate probabilities from repeated outcome counts and invert the resulting linear equations.Source: B. Standard tomography · p. 2
- Linear-inversion methods remain useful as a first stage or in the large-sample limit, despite more sophisticated statistical methods offering better finite-sample accuracy.Their existence also guarantees efficient, reasonably accurate protocols for standard tomography.Source: B. Standard tomography · p. 2
- Standard process tomography assumes known test states and measurement outcomes, although these are often generated by the same gates being characterized.Those assumptions define an absolute reference frame for the system's state space.Source: B. Standard tomography · p. 2
II. GATE SET TOMOGRAPHY
Gate set tomography treats the device as a black box whose initialization, measurement, and gate operations are all inferred from data. It aims to identify a complete gate-set description without assuming that particular operations prepare or measure ideal states.
II. GATE SET TOMOGRAPHY
- GST infers the effects of the initialization, measurement, and gate buttons from data, without assuming that any operation prepares a particular state or measures a particular basis.If the operations behave that way, the data are expected to reveal it.Source: §II · p. 3
- The black-box model assumes known Hilbert-space dimension, repeatable re-preparation by the initialization button, a two-outcome POVM measurement, and Markovian control.The stated assumptions can be relaxed at some cost to make GST more robust.Source: §II · p. 3
- A gate set is the complete black-box description GST seeks to identify from experiments.The device is modeled through its initialization state, measurement effect, and gate maps.Source: §II · p. 3
A. Experiments, data, and inference in GST
GST gathers repeated outcomes from sequences of gates applied between initialization and measurement, then uses the resulting probabilities to infer the gate set. Sequence experiments are intended to be sufficient to identify that set.
A. Experiments, data, and inference in GST
- Each GST experiment initializes the system, applies a sequence of gates, and records a measurement outcome; sequences correspond to circuits without assuming gates commute.The account treats sequences as experiments indexed by the gate sequence and records counts for each sequence performed.Source: A. Experiments, data, and inference in… · p. 3
- The probability of an outcome for a gate sequence is computed from the initialized state, the ordered gate maps, and the measurement effect.Observed counts provide information about that single probability and thus about the gate-set parameters.Source: A. Experiments, data, and inference in… · p. 3
- A sufficient collection of distinct experiments should identify the gate set, whose parameter count grows with Hilbert-space dimension and the number of gates.For dimension d, the state requires d^2 − 1 parameters, the measurement effect d^2, and each gate d^2(d^2 − 1).Source: A. Experiments, data, and inference in… · p. 3
B. The gauge
Gate-set descriptions related by an invertible change of representation predict identical observable probabilities, leaving some parameters unobservable. Positivity constraints complicate this gauge freedom, and this work does not impose complete positivity on estimated gates.
B. The gauge
- An invertible superoperator transforms a gate set into a gauge-equivalent representation with identical observable probabilities.The gauge group is SL(d^2), with scalar multiples of a transformation acting identically.Source: B. The gauge · p. 3
- Gauge freedom leaves approximately d^2 − 1 parameters redundant and unobservable, corresponding to an arbitrary reference frame rather than observable dynamics.Gate sets that look different in their state and gate representations can be experimentally indistinguishable.Source: B. The gauge · p. 4
- The paper lacks a satisfying gauge-invariant normal form or a well-motivated gauge-invariant fidelity measure, so it compares estimates with targets only after numerical gauge optimization.The estimate is obtained without considering the target; only afterward is its representation varied to minimize discrepancy.Source: B. The gauge · p. 4
C. Positivity
- Gauge transformations need not preserve conventional positivity constraints, and directly requiring each gate to be completely positive truncates gauge freedom.For extremal gate sets, complete positivity reduces the gauge group to SU(d); for noisy sets, its consequences are more complicated.Source: C. Positivity · p. 4
- This work does not impose complete positivity, instead allowing arbitrary estimated gate matrices and relying on consistency with data to ensure positive predicted probabilities.Careful implementation of positivity constraints is identified as future work.Source: C. Positivity · p. 4
D. Practical inference of the gate set
The section motivates maximum-likelihood estimation as a robust alternative to reconstructing gates from separately estimated probabilities. In GST, however, the likelihood can be nonconvex and have multiple local maxima, making prior-dependent optimization risky.
D. Practical inference of the gate set
- Maximum-likelihood estimation selects the gate set that maximizes a likelihood built from the observed counts and predicted experiment probabilities.Directly estimating probabilities and reconstructing the gate set is impractical because observed frequencies rarely equal their expected probabilities exactly.Source: D. Practical inference of the gate set · pp. 4–5
- GST likelihoods are generally not log-convex or guaranteed to have a unique local maximum because experiment probabilities depend nonlinearly on gate-set parameters.The probabilities are polynomials whose degree can grow with the maximum number of times a gate appears in an experiment.Source: D. Practical inference of the gate set · p. 5
- Earlier methods expanded the likelihood near target gates, risking circular estimates when prior knowledge is inaccurate and optimization finds a nearby local maximum.The authors propose a robust closed-form estimator as a direct alternative or a starting point for likelihood maximization.Source: D. Practical inference of the gate set · p. 5
III. LINEAR GATE SET TOMOGRAPHY
Linear inversion offers a closed-form route to tomography, but gate-set estimation is harder because its event probabilities depend nonlinearly on the unknown gates. LGST uses experimentally measured sequence probabilities and a Gram-matrix-style construction to estimate the gate set up to gauge.
III. LINEAR GATE SET TOMOGRAPHY
- In overcomplete linear inversion, least-squares weights matter because they determine which conflicting observations dominate, and choosing them can require likelihood estimation.The best weighted least-squares fit is the argmax of a Gaussian approximation to the likelihood.Source: §III · p. 5
- Gate-set likelihoods may be non-unimodal because event probabilities depend nonlinearly on gate-set parameters, unlike the linear setting of state tomography.This makes the feasibility of direct likelihood-based GST less obvious than for state tomography.Source: §III · p. 5
- LGST provides a closed-form gate-set estimate and a useful starting point for likelihood optimization, rather than serving as the final accuracy-optimal estimator.Its estimate is obtained through a simple linear-inversion method using a Gram-matrix technique.Source: §III · p. 5
A. Derivation of the LGST algorithm
- For each gate X, LGST measures probabilities for sequences F_j X F_k and arranges them into a matrix for linear inversion.The measured probabilities correspond to experimentally observable events.Source: A. Derivation of the LGST algorithm · p. 5
- When the identity-sequence matrix is invertible, the resulting gate estimates are gauge-equivalent to the unknown gates.The construction also estimates the state and measurement effect from corresponding vectors.Source: A. Derivation of the LGST algorithm · pp. 5–6
B. How to implement LGST
LGST is implemented by repeating a set of short gate-sequence experiments, estimating their probabilities, and assembling those probabilities into matrices and vectors. Rank checks diagnose incomplete information and guide changes to the fiducial sequences.
B. How to implement LGST
- LGST gathers statistics from repeated experiments, arranges estimated probabilities into matrices, and constructs estimates of the state, effect, and gates.The experimental probabilities are obtained from the chosen sequence set.Source: B. How to implement LGST · p. 6
- A rank-deficient identity matrix indicates that the state or measurement effects are informationally incomplete; replacing fiducial sequences may restore full rank.Persistent rank deficiency indicates the gate set cannot generate an informationally complete set and requires hardware-level intervention.Source: B. How to implement LGST · p. 6
- A full-rank matrix with small eigenvalues signals marginal informational completeness, where statistical fluctuations are amplified by inversion.The authors address this by adding sequences and removing those whose removal maximizes the minimum eigenvalue.Source: B. How to implement LGST · p. 6
IV. IMPLEMENTATION OF GST ON A TRAPPED-ION QUBIT
The experiment applies GST to four gates generating the single-qubit Clifford gates of a trapped-ion qubit. Training data produced estimates, while hidden testing experiments were reserved to score predictive accuracy objectively.
IV. IMPLEMENTATION OF GST ON A TRAPPED-ION QUBIT
- Four gates generating the single-qubit Clifford gates were characterized, and the authors did not attempt to minimize their errors because the experiment evaluated GST.The analysis nevertheless found the gates accurate enough that standard tomography would have worked fairly well, a conclusion supported by GST’s robust framework.Source: §IV · p. 7
- Training sequences generated the tomographic estimates, while hidden testing sequences objectively scored how well each tomography method predicted experiments.Training and testing were interleaved to reduce drift effects during data collection, and gate pulse amplitude was recalibrated periodically.Source: §IV · p. 7
- The LGST estimate used 84 distinct experiments repeated 1,900 times each, then underwent numerical gauge optimization against target gates.Table I presents the gauge-optimized LGST estimates adjacent to the intended target gates.Source: Table I · p. 7, §IV · p. 7
- Maximum-likelihood estimates were reported for short datasets of 85 sequences and long datasets of 1,066 sequences, alongside target gates.The supplied table caption specifies these dataset scopes but does not state which estimate performs better.Source: Table II · p. 8
V. IMPROVING GST WITH MAXIMUM LIKELIHOOD
Maximum-likelihood GST can use nonlinear and overcomplete data for accuracy, but its likelihood may be nonconvex. LGST provides a reliable closed-form starting estimate for ML refinement, and long-sequence training substantially improves long-test-sequence predictions.
V. IMPROVING GST WITH MAXIMUM LIKELIHOOD
- GST likelihoods can be nonconvex, but LGST supplies a nearby starting point for gradient ascent; ML then refines it with nonlinear and overcomplete data.The authors expect gradient ascent from the LGST estimate to find the global maximum; this is their rationale for the hybrid approach.Source: §V · p. 8
- LGST is a closed-form, linear-inversion protocol that avoids likelihood local maxima, while hybrid LGST-plus-ML recovers accuracy beyond the bare estimate.The authors report LGST’s reliability comes at the price of lower accuracy than ML.Source: §V · p. 8
- The long-data ML fit used 1066 distinct sequences, each repeated 1900 times; the experiment choices were described as somewhat arbitrary.The long dataset included sequences of powers of gates, analyzed using maximum likelihood.Source: §V · pp. 8–9
- ≈0.01 score per count: long-training ML remains accurate on length-100 test strings, while bare LGST rapidly becomes inaccurate beyond L=5.Figure 3 reports score per count; lower scores are better.Source: Figure 3 · p. 9
VI. QUANTIFYING ACCURACY OBJECTIVELY WITH SCORED TESTS
Because tomography estimates are theories intended to predict future experiments, their accuracy should be tested on data not used to fit them. The paper scores predictions on independent test experiments using observable probabilities, avoiding reliance on an unknown true process or gauge-dependent fidelity.
VI. QUANTIFYING ACCURACY OBJECTIVELY WITH SCORED TESTS
- Tomographic estimates are scored on held-out testing experiments using predicted outcome probabilities, rather than fit to training data.The score uses the logarithmic scoring rule and is based on observable probabilities that are gauge-invariant.Source: §VI · p. 9
- For each binary test outcome, the estimate receives a negative-log-probability score, which is aggregated over the testing data.The experiments have two outcomes, labeled + and −.Source: §VI · p. 9
- The score is both the relative entropy between predicted probabilities and empirical frequencies and the estimate’s loglikelihood on testing data.Lower scores are better; the score is renormalized by subtracting the minimum attributable to the data’s entropy.Source: §VI · p. 9
- A zero predicted probability receives an infinite penalty, while negative probabilities make the scoring formalism fail.The work did not impose positivity, and some estimates predicted negative probabilities.Source: §VI · pp. 9–10
1. The target gates themselves,
The comparison tests estimates on held-out sequences of increasing length, including long and nonuniform gate strings. Long-training ML predicts the longest tests well, whereas bare LGST’s predictive accuracy degrades beyond the lengths of its training data.
1. The target gates themselves,
- The long dataset added sequences with gate powers up to p=128 and corresponding experiments ending with an additional G4 gate.It contained 1066 sequences, each repeated 1900 times, to probe long-sequence training.Source: §1 · p. 10
- All estimates’ scores rise with test-sequence length, and shorter sequences are easier to predict.Figure 3 averages score per count over the ten test sequences at each length.Source: §1 · p. 11
- For L≤5, all three tomographic estimates predict fairly well, while target gates perform poorly even on short sequences, mostly due to SPAM error.Maximum-likelihood methods outperform LGST even at L=4 and L=5.Source: §1 · p. 11
- ∼0.02 score per count at L=100: the best ML estimate retains predictive power on long test sequences.The long dataset was intended to probe the utility of long gate sequences for accuracy and the feasibility of GST estimation on them.Source: §1 · pp. 10–11
- LGST works well on strings matching its training length but degrades rapidly beyond L=3; its key role is to seed ML, not provide an optimal estimate.The authors suggest overcomplete data and appropriately weighted least-squares fitting might improve LGST accuracy.Source: §1 · p. 11
VII. CONCLUSIONS
GST is presented as a reliable, self-consistent framework for characterizing quantum gates without precalibrated reference frames. The authors report that it can combine an LGST starting estimate with maximum-likelihood estimation and assess estimates by predicting future data.
VII. CONCLUSIONS
- LGST provides a closed-form first estimate that makes GST robust against likelihood local maxima while allowing maximum-likelihood methods to achieve high accuracy.The authors describe this as a way to obtain a closed-form approximation to the true gates before local maximum-likelihood estimation.Source: §VII · p. 11
- GST’s scoring protocol uses testing experiments and observable probabilities to assess estimates without relying on target gates.The authors describe prediction of future data as the estimate’s fundamental job.Source: §VII · p. 11
- GST estimates are reported to be good predictors, and the proposed scoring protocol evaluates them by their ability to predict future data.The protocol avoids judging estimates against potentially incorrect targets or similarly biased estimates.Source: §VII · pp. 11–12
- GST estimates the gate set self-consistently and treats gates as critical because repeated gate applications generate many observable probabilities.The authors state that 2^L distinct observable probabilities can be generated using only 2 distinct gates, without adding extra parameters to estimate.Source: §VII · p. 12
- The authors identify an SL(d^2) gauge as a necessary price of GST and describe it as a significant complication.The passage calls this gauge intriguing and pernicious but does not elaborate further here.Source: §VII · p. 12