PaperlayerView exact revision on arXiv

Source-linked AI summary

Introduction to Quantum Gate Set Tomography

Daniel Greenbaum

arXiv:1509.02921v1quant-ph

TL;DR

The paper addresses the need for accurate, full characterization of quantum gates when SPAM and coherent errors make QPT unreliable. It reviews and guides the implementation of GST, then compares GST and QPT using simulated single-qubit data. GST is accurate within sampling limits in the presence of SPAM errors, while QPT typically overestimates gate error and poorly estimates coherent errors near QEC-relevant regimes.

  • Problem

    QPT assumes ideal state preparation and measurement, limiting accurate gate characterization when SPAM errors are present, especially for fault-tolerant quantum information processing.

  • Method

    The paper provides a self-contained review and implementation guide for GST, including mathematical background, estimation procedures, and simulated single-qubit comparisons with QPT.

  • Results

    GST is accurate within sampling error with SPAM errors, whereas QPT typically overestimates gate error and provides less access to systematic error information.

  • Takeaways & Limitations

    GST offers a robust tool for full gate characterization and qualitative diagnosis of systematic errors in regimes relevant to quantum error correction.

  • Takeaways & Limitations

    Estimating GST uncertainty and obtaining the accuracy required for QEC may impose substantial resource demands, particularly for two-qubit gates.

Abstract

from arXiv · show

Quantum gate set tomography (GST) has emerged as a promising method for the full characterization of quantum logic gates. In contrast to quantum process tomography (QPT), GST self-consistently and correctly accounts for state preparation and measurement (SPAM) errors. It therefore provides significantly more accurate estimates than QPT as gate fidelities increase into the fault-tolerant regime. We give a detailed review of GST and provide a self-contained guide to its implementation. The method is presented in a step-by-step fashion and relevant mathematical background material is included. Our goal is to demonstrate the utility of GST as both an accurate characterization technique and a simple and effective diagnostic tool. As an illustration, we compare the output of GST and QPT using simulated example data for a single qubit. In agreement with the original literature, we find that coherent errors are poorly estimated by QPT near quantum error correction thresholds, while GST is accurate in this regime.

Introduction

The paper motivates GST as a self-consistent alternative for accurate, diagnostically useful qubit characterization, especially when SPAM and coherent errors matter near fault-tolerant thresholds. It reviews GST’s mathematical foundations, implementation, and comparison with QPT using simulated single-qubit data.

  • Motivation: Accurate qubit characterization is needed to identify control, crosstalk, and environmental errors and assess whether gates meet QEC thresholds.Full process characterization also provides gate-error information and quality metrics.
  • Related methods: GST provides richer error information than randomized benchmarking, although RB is more scalable and GST requires substantially more experiments and processing.The paper reports approximately 80 experiments for one qubit and over 4,000 for two qubits, compared with 16 and 256 for single-gate QPT.
  • GST versus QPT: QPT can confuse systematic gate errors and becomes less reliable as gate quality improves, while GST is designed to resolve these diagnostic limitations.For example, QPT may not distinguish an over-rotation on one gate from the same error distributed across all gates.
  • GST versus QPT: QPT becomes unreliable with SPAM errors because it assumes ideal state preparation and measurement, whereas GST models these operations self-consistently.This limitation is especially important when intrinsic SPAM errors are large relative to gate errors.
  • Paper scope: The document offers a practical, self-contained single-qubit guide covering quantum-map representations, tomography, LGST, and maximum-likelihood estimation.LGST is presented as both an initialization method for nonlinear MLE and a source of gate-set information in its own right.
  • Illustration: Simulated data incorporating coherent and incoherent errors and finite sampling noise are used to compare QPT and GST.The comparison targets the methods’ performance under realistic error features.

2.2 Superoperator formalism

The superoperator formalism represents quantum states as Hilbert-Schmidt vectors and quantum operations as matrices, making map composition and experimental probabilities straightforward matrix expressions.

  • Representation: Density operators become vectors in a d^2-dimensional Hilbert-Schmidt space, while quantum operations become d^2 × d^2 matrices.The formalism uses the Hilbert-Schmidt inner product to define the vector representation.
  • Measurements: POVMs represent measurements through p = Tr{Eρ}, with positive semidefinite measurement operators satisfying a completeness relation.The paper mainly uses projective measurements as a special case.
  • Pauli basis: Using a normalized Pauli basis provides a convenient coordinate system for states and operations and identifies PTM entries with superoperator matrix elements.Rescaling avoids repeated factors of d.
  • Composition: Composition of quantum maps is represented by matrix multiplication in the superoperator formalism.This follows from applying successive maps to an arbitrary state.
  • Experimental probabilities: An experiment that prepares ρ, applies Λ, and measures E has expected outcome p = Tr{EΛ(ρ)} = ⟨⟨E|RΛ|ρ⟩⟩.The expression connects laboratory measurements to the state, map, and measurement representations.

2.3 Physicality constraints

Physical quantum maps must preserve probabilities and remain positive on system–environment states; the paper translates these requirements into constraints on process representations.

  • CPTP requirements: Complete positivity requires nonnegative probabilities for every initial state of the universe, while trace preservation requires conservation of total probability.Together these conditions define physical CPTP maps.
  • χ representation: The process-matrix constraints follow from the general χ-representation of a quantum map and the CPTP requirements.The paper explicitly derives these constraints from complete positivity and trace preservation.
  • χ representation: In the χ representation, complete positivity implies χ is positive semidefinite, and trace preservation yields the completeness condition.The derivation uses probabilities of arbitrary pure-state transitions and the Kraus representation.
  • PTM representation: In the Pauli transfer matrix representation, trace preservation has a direct constraint, whereas complete positivity is imposed through the associated Choi-Jamiolkowski matrix.The CP condition is ρΛ ≽ 0.

Derivation of GST

The paper introduces GST as a self-consistent extension of QPT and organizes the surrounding material to support its derivation and implementation.

  • Derivation of GST: The chapter derives GST and relevant data-analysis techniques before presenting experimental and analysis protocols step by step.Readers focused on implementation are directed toward the following chapter.
  • Derivation of GST: GST is presented as a self-consistent extension of quantum process tomography, which itself developed from quantum state tomography.The chapter therefore begins by reviewing QST and QPT.

3.1 Quantum state tomography

Quantum state tomography estimates an unknown state from measurement probabilities, using linear inversion when the measurement matrix is invertible. Sample averages converge to the true probabilities as the number of measurements increases.

  • QST characterizes an unknown state ρ by measuring its components or probabilities in a chosen operator basis.The measurement operators must span the Hilbert-Schmidt space.
  • Given known A_jk, linear inversion estimates the state as |ρ̂⟩⟩ = A^−1|m⟩⟩ when A is invertible.The measurement operators must be chosen so that A has an inverse.
  • The sample average m_j estimates probability p_j from N single-shot measurements with binary outcomes.Its expected value is p_j, and its variance is p_j(1 − p_j)/N.
  • As N →∞, the sample average m_j approaches the true probability p_j.
  • When |E_j⟩⟩ = |j⟩⟩, estimation requires no matrix inversion and gives |ρ̂⟩⟩ = |m⟩⟩.

3.2 Quantum process tomography

Quantum process tomography reconstructs a gate from probabilities measured with assumed known states and measurements. Gate set tomography instead estimates SPAM gates self-consistently, addressing errors in those assumptions while requiring only one initial state and measurement.

  • 3.2 Quantum process tomography: QPT characterizes a gate G by measuring its process matrix χ or PTM through d^4 probabilities.
  • 3.2 Quantum process tomography: QPT reconstructs the PTM from gate-sandwiched measurements using experimenter-specified states and POVMs.The states and measurements form complete bases, enabling linear inversion or least-squares estimation.
  • 3.2 Quantum process tomography: Overcomplete state and measurement sets make S rectangular, so ordinary least squares uses S†S rather than directly inverting S.
  • 3.2 Quantum process tomography: QST and QPT estimates can be faulty when the gates preparing initial states and final measurements contain systematic SPAM errors.
  • 3.3 Gate set tomography: GST solves the self-consistency problem by including SPAM gates in the gate set being estimated.Its goal is complete characterization of an unknown set of gates and states.
  • 3.3 Gate set tomography: GST requires only one prepared initial state and one implemented measurement because the SPAM gates are included in the gate set.This matches common experimental choices such as a qubit ground state and a Z-basis measurement.

3.4 Linear inversion GST

Linear-inversion GST constructs gate estimates from experimentally measurable expectation values using SPAM gate strings and Gram-matrix relations, but the estimates retain gauge freedom and may be unphysical. Gauge optimization aligns the estimate with a chosen target while preserving experimental consistency.

  • Linear inversion: LGST is useful for diagnosing gate errors and initializing later optimization, but its unconstrained estimates can be unphysical and cannot by themselves provide gate-quality metrics.When the LGST estimate is physical, it coincides with the estimate found by maximum-likelihood estimation.
  • SPAM gates: LGST begins with SPAM gate strings that generate complete sets of initial and final states from an unknown state and measurement.For a single qubit, the generated states and measurements must span the Bloch sphere.
  • Measured quantities: The experimentally measurable quantities p_ijk are expectation values associated with SPAM gates F_i, F_j and gate G_k.In the Pauli-transfer representation, these measurements satisfy p_ijk = (AG_kB)_ij.
  • Gram matrix: The null-gate experiment supplies a Gram matrix, enabling LGST to estimate the gate set up to an unobservable similarity transformation.The Gram matrix must be invertible; small or zero eigenvalues indicate that the implemented SPAM gates are not linearly independent enough for estimation.
  • Gauge freedom: Gauge transformations leave all measured experiments unchanged, so GST estimates are identifiable only up to similarity transformation and must be compared in a selected gauge.A target gate set can define the gauge used to compare estimated gates with intended gates.
  • Gauge optimization: Gauge optimization selects the gate set closest to a target while remaining consistent with the experiments and allowable gauge freedom.The nonlinear optimization may have multiple minima and requires a specified starting point.

3.5 Maximum likelihood estimation

Maximum-likelihood GST fits a physical gate-set model to experimental probabilities, using parameterizations and constrained nonlinear optimization to obtain self-consistent estimates. The process-matrix and PTM formulations trade objective-function order against positive-semidefiniteness constraints.

  • Maximum-likelihood estimation: MLE fits a theoretical probability model to measurements and finds the most likely gates, state, and measurement that produced the data.The estimated probabilities are constrained to be physical, and the resulting gate set includes state-preparation and measurement estimates.
  • Parameterization: Parameterizing gates, states, and measurements by a shared vector makes each estimated probability a fifth- or tenth-order homogeneous function.Linear parameterizations produce order 5; quadratic parameterizations produce order 10.
  • Objective function: The Gaussian approximation converts maximum-likelihood estimation into weighted least-squares using a sampling variance based on measured probabilities.The variance is σ2 = m(1 −m)/n in the numerical implementation.
  • Optimization: The resulting optimization is non-convex because estimated probabilities can be tenth- to twentieth-order in the parameters, so local methods require a good starting point.Linear inversion supplies a useful initialization even though it may produce unphysical estimates and cannot exploit overcomplete data.
  • Physicality constraints: Physicality is enforced through constraints including positive semidefiniteness, trace preservation, unit state trace, and positivity of the complementary measurement operator.The process matrix uses Cholesky parameterization, while state and measurement constraints are imposed during optimization.
  • Pauli transfer matrix optimization: The PTM formulation lowers the objective from twentieth- to tenth-order but introduces positive-semidefiniteness constraints on each gate’s associated matrix.This tradeoff naturally suggests a semidefinite-program formulation.

Implementing GST

Implementing GST requires solving a nonlinear estimation problem while enforcing self-consistency across gates, state preparation, and measurement. The paper demonstrates a practical workflow with simple nonlinear optimization and simulated single-qubit comparisons against QPT.

  • Implementing GST: GST estimates a complete gate set from measured probabilities that jointly depend on preparation, measurement, and gate sequences.QPT becomes inadequate for self-consistent estimation because including SPAM gates makes the gate-sequence dependence nonlinear.
  • Optimization: For sufficiently repeated experiments, GST estimation can use maximum likelihood with a quadratic objective in estimated probabilities, but gate sequences make the parameter optimization nonlinear.When each sequence contains a single gate, as in QPT, the objective is quadratic in gate parameters.
  • Numerical implementation: The implementation uses full nonlinear Matlab optimization with reasonable, unoptimized settings to illustrate GST’s main features.The authors rely on approximate and iterative methods rather than procedures guaranteed to locate the global optimum.
  • Numerical results: Simulated single-qubit data compare maximum-likelihood GST and QPT across coherent error, incoherent error, and sampling-noise levels.The comparison is designed to assess estimation performance under several error regimes.
  • Numerical results: Near quantum-error-correction thresholds, QPT poorly estimates coherent errors while GST remains accurate in the reported simulations.The result corroborates earlier literature cited by the authors.

4.1 Experimental protocol

The GST experimental protocol constructs a basis of starting states and measurements from SPAM gates, applies gate sequences between them, and estimates expectation values through repeated binary measurements. The procedure is repeated across all SPAM-gate and gate-set combinations.

  • Experimental protocol: For a single qubit, the protocol uses at least four SPAM gates to construct a complete basis of starting states and measurements.The paper takes N = 4 for simplicity, with each SPAM gate composed of gates from the gate set.
  • Experimental protocol: Each experiment initializes the qubit, applies Fi ◦Gk ◦Fj, measures the POVM E, and records whether the desired outcome occurs.The SPAM gates themselves are implemented as sequences of gates from the gate set.
  • Repeated measurements: Repeating each experiment n times, typically 1,000–10,000, produces a binary sample average mijk estimating pijk.The estimator has mean pijk and variance pijk(1 −pijk)/n.
  • Data collection: The experiment repeats the initialization, sequence, measurement, and averaging steps for every i, j and k combination.Optional direct measurements of ⟨⟨E|Fi|ρ ⟩⟩ can provide independent data when available.

4.2 Organizing and verifying the data

GST data are organized into matrices and checked through the Gram matrix before inversion. The SPAM-gate basis must be sufficiently linearly independent, because small Gram-matrix eigenvalues amplify sampling error.

  • Organizing the data: The measured quantities are organized into matrices indexed by the SPAM-gate and gate-set labels.The matrices represent the experimentally measured versions of the corresponding GST quantities.
  • Verifying the data: Before further processing, the Gram matrix gij = ⟨⟨E|FiFj|ρ ⟩⟩ must be nonsingular so it can be inverted for the GST estimate.The smallest-magnitude eigenvalue should be as large as possible.
  • Verifying the data: Sampling error in the GST estimate scales roughly with the inverse of the Gram matrix’s smallest eigenvalue.Thus, poorly conditioned Gram matrices make the estimate more sensitive to sampling noise.
  • Verifying the data: Very small Gram-matrix eigenvalues indicate that the SPAM gates are only marginally linearly independent and highly overlapping.The experimenter should adjust the experiment to make the SPAM gates more orthogonal, then remeasure and recheck the eigenvalues.

4.3 Linear inversion

The section develops linear inversion GST as an accessible diagnostic and initialization procedure, then situates it within physical, single-qubit maximum-likelihood estimation. LGST is useful for quick qualitative checks and as a starting point for MLE, but it does not generally enforce physicality.

  • Linear inversion: Applying the inverse experimental Gram matrix to the data matrices yields estimates for the gates and states.The null-gate LGST estimate is exactly the identity because the Gram matrix is the null-gate data matrix.
  • Gauge optimization: Gauge optimization transforms the LGST estimate toward the intended target gate set before comparing the final estimate with the target.The optimized gauge matrix is applied to the estimated gates, state, and measurement.
  • Diagnostics: LGST is easy to implement and useful for quick diagnostics, but its information is qualitative because the estimate is not generally physical.Large errors can still be detected, while physicality constraints require a separate procedure.
  • Maximum-likelihood estimation: LGST provides a starting point for MLE by selecting the closest physical gate set, with a distance metric such as the trace norm.MLE is preferred because the starting point must be physical and the closest-physical-set approach is suboptimal.
  • Simulation setup: The simulated study models a single qubit with orthogonal X and Y rotations, faulty preparation and Z-basis measurement, coherent Y over-rotations, depolarizing noise, and sampling noise.GST and QPT are compared using maximum-likelihood estimates; the GST estimate is initialized from LGST and QPT from the target gate set.

Example 1: Over-rotation (coherent) error

The coherent-error example compares ML-GST and ML-QPT for a single faulty Yπ/2 gate across over-rotation strengths. GST identifies the faulty gate and remains accurate near QEC-relevant error rates, whereas QPT spreads the error across gates and becomes much less reliable.

  • Estimation behavior: QPT attributes error to all gates because the faulty gate appears in state-preparation-and-measurement sequences, whereas only Yπ/2 is faulty.GST correctly identifies the gate containing the over-rotation.
  • Estimation behavior: GST estimation error is flat as gate error varies, reflecting the optimizer’s numerical threshold in this simulation.The stated tolerance corresponds to an estimation error of approximately 10^-7.
  • Setup: 8.5×10^-4 gate infidelity corresponds to a 4° Yπ/2 over-rotation used for the representative PTM comparison.Figure 4.1 relates gate error to the over-rotation angle, while Figures 4.2 and 4.3 use this selected error.
  • Results: In the QEC-relevant regime of gate error 10^-3–10^-2, GST is several orders of magnitude more accurate than QPT.The comparison is based on fidelity-derived gate and estimation errors for physical estimates.
  • PTM diagnostics: The PTM comparison shows GST and QPT both detect the faulty Yπ/2 gate, but QPT additionally assigns rotations to the other gates.The error-difference PTMs show GST errors about 1% of QPT errors and invisible on the displayed scale.
  • Comparison with incoherent error: For depolarizing noise, QPT and GST have about the same overall estimation error, but GST estimates the non-zero PTM coefficients and depolarizing parameter p more accurately.The non-zero coefficients equal 1−p, illustrating that a single quality metric can miss differences in estimation quality.

Example 3: Intrinsic SPAM error

The intrinsic-SPAM example shows that QPT attributes initial-state depolarization to gates, whereas GST remains insensitive to this SPAM error. Under sampling noise, GST can still identify coherent over-rotations below the sampling-error crossover.

  • Metric choice: The analysis uses spectral norm for depolarizing-noise estimation errors because physicality constraints were not satisfied tightly enough for infidelity.The constraint violation was small but sufficient to invalidate the use of infidelity in that example.
  • Intrinsic SPAM error: For depolarizing noise on gates, GST and QPT estimation errors are comparable, but QPT estimates the depolarizing parameter less accurately.The depolarizing gate error is E(actual, ideal) = 8.5 × 10^-4.
  • Intrinsic SPAM error: QPT estimation error grows linearly with initial-state depolarization and is almost equal to the state error.Because QPT assumes an ideal initial state, it attributes the noise to the gates.
  • Sampling noise: Under sampling noise with NSamples = 10000 and intrinsic SPAM error ⟨⟨E|ρ⟩⟩ = 0.01, both methods remain flat in estimation error until gate error exceeds 0.01.QPT is limited by intrinsic SPAM, whereas GST is not.
  • Sampling noise: At gate error near 10^-3, GST still detects the over-rotation because its non-zero PTM entries exceed the average sampling-noise entry magnitude.This gate error is below the 0.01 crossover point.
  • Intrinsic SPAM error: QPT incorrectly assigns initial-state depolarization to all gates, while GST avoids this misattribution.The actual gate set has no gate error; the depolarizing noise is applied only to the initial state.
Loading 1509.02921v1…