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Self-Consistent Quantum Process Tomography
Seth T. Merkel, Jay M. Gambetta, John A. Smolin, S. Poletto, A. D. Córcoles, B. R. Johnson, Colm A. Ryan, M. Steffen
TL;DR
Standard quantum process tomography can be grossly inaccurate when its preparation and measurement gates have systematic errors, because those SPAM errors invalidate the assumed likelihood model. The paper instead reconstructs an entire gate library self-consistently by incorporating uncertainty in SPAM gates and linearizing around target gates. In simulation and experiments, this approach produces more trustworthy, generally higher-fidelity estimates than standard QPT while adding only polynomial classical overhead.
Problem
Systematic errors in the gates used for state preparation and measurement can make standard QPT fidelity estimates very poor and cannot be fully corrected by more sampling.
Source: INTRODUCTION · p. 1, THE TROUBLE WITH TOMOGRAPHY · p. 3Method
The method modifies QPT's likelihood function to estimate an entire gate library self-consistently without assuming a detailed SPAM error model, using linearization around the ideal gates.
Source: SELF-CONSISTENT TOMOGRAPHY · p. 6Results
Self-consistent tomography outperforms standard QPT in simulation and yields more trustworthy, generally higher-fidelity estimates in simulation and experiment.
Source: INTRODUCTION · p. 1, CONCLUSION · pp. 8–9Takeaways & Limitations
The method preserves full gate characterization while accounting for SPAM uncertainty, with the difference from standard QPT expected to grow at lower gate errors.
Source: CONCLUSION · pp. 8–9The method adds nontrivial polynomial overhead, becomes numerically limited for larger systems, and requires expensive diagonal-frame maximization.
Source: SELF-CONSISTENT TOMOGRAPHY · p. 8
Abstract
from arXiv · showhide
Quantum process tomography is a necessary tool for verifying quantum gates and diagnosing faults in architectures and gate design. We show that the standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography can not be fully corrected through oversampling or by performing a larger set of experiments. We present an alternative method for tomography to reconstruct an entire library of gates in a self-consistent manner. The essential ingredient is to define a likelihood function that assumes nothing about the gates used for preparation and measurement. In order to make the resulting optimization tractable we linearize about the target, a reasonable approximation when benchmarking a quantum computer as opposed to probing a black-box function.
INTRODUCTION
Quantum process tomography is needed for full gate characterization, but systematic errors in the preparation and measurement gates can make standard estimates unreliable. The paper proposes self-consistent reconstruction of an entire gate library while treating those SPAM gates as uncertain.
Motivation
- QPT provides full process information needed to diagnose and improve quantum gates beyond a single fidelity value.General maps require O(24n) parameters, so tomography scales exponentially with qubit number but is commonly applied to small subsystems.Source: INTRODUCTION · p. 1
The SPAM problem
- Fixed preparation states and measurement operators are themselves generated by imperfect gates, so systematic SPAM errors can be comparable to the gate errors being estimated.These errors cannot be corrected simply by collecting more measurement statistics or enforcing physicality.Source: INTRODUCTION · p. 1
- Standard QPT can produce very poor fidelity estimates as coherent SPAM errors decrease, even while randomized benchmarking compensates for SPAM in fidelity estimation.Randomized benchmarking does not provide the full tomographic information needed to diagnose and correct gates.Source: INTRODUCTION · p. 1
Approach
- Increasing the number of SPAM gates does not generally improve tomography for unitary errors, although it helps for uniform errors such as a global frame transformation.The proposed alternative places SPAM gates and interrogated gates on the same footing and reconstructs the library self-consistently.Source: INTRODUCTION · p. 1
- Self-consistent reconstruction assumes little about gate-error models and has polynomially equivalent measurement and post-processing cost to standard QPT on each library gate.The paper contrasts this model-light approach with self-correcting state tomography and bootstrap methods.Source: INTRODUCTION · pp. 1–2
THE TROUBLE WITH TOMOGRAPHY
Standard QPT estimates processes from measured preparations and observables, but systematic SPAM errors alter the likelihood model and can make reconstructions inaccurate or unphysical. Simulations show that increasing sampling reduces stochastic error only to a systematic floor, with coherent errors producing especially poor scaling.
QPT formulation
- QPT represents a process as a CPTP map and estimates its vectorized transfer matrix by linear inversion from experiments defined by prepared states, measured observables, and expectation values.The Choi matrix enforces complete positivity, while the Pauli-transfer representation makes composition matrix multiplication.Source: THE TROUBLE WITH TOMOGRAPHY · p. 2
Likelihood limitation
- Gaussian measurement noise decreases with repetitions, but maximum-likelihood estimation can only enforce physicality; it cannot correct a likelihood function built from mischaracterized SPAM gates.With systematic SPAM errors, the preparation and measurement vectors are transformed by unknown error maps, changing the experiment matrix used in reconstruction.Source: THE TROUBLE WITH TOMOGRAPHY · pp. 2–3
SPAM-induced reconstruction error
- For large measurement-noise powers, the identity-reconstruction error decreases exponentially as repetitions increase, then reaches a floor set by systematic SPAM error.The estimate is compared with the identity using diamond norm distance, without imposing the CPTP constraint.Source: THE TROUBLE WITH TOMOGRAPHY · p. 3, Figure 1 · p. 3
Oversampling
- Oversampling with larger unitary gate libraries can speed convergence but does not change the systematic estimation limit.The libraries include tetrahedral and cardinal-state sets plus 12- and 24-element Clifford 2-designs.Source: THE TROUBLE WITH TOMOGRAPHY · pp. 3–4
Systematic-error models
- Coherent SPAM errors make the reconstruction-error ratio grow exponentially as individual gate errors decrease, whereas incoherent errors produce proportional tomography error.The comparison uses the closest physical estimate under a flat metric and includes over-rotation, detuning, amplitude damping, dephasing, and depolarizing models.Source: THE TROUBLE WITH TOMOGRAPHY · p. 4, Figure 2 · p. 4
Benchmarking implications
- For fault-tolerant target errors of 10^-3 to 10^-5, coherent SPAM can make QPT report roughly 10^-2 to 10^-3, while reported 10^-5 errors correspond to physical errors below 10^-8.At typical current-experiment noise powers, SPAM errors are expected to dominate estimates below about one percent.Source: THE TROUBLE WITH TOMOGRAPHY · p. 4
QPT WITH OVERSAMPLING
Oversampling by enlarging the gate library does not generally remove SPAM-induced tomography errors, although unitary 2-designs can cancel some uniform errors. Fidelity and diamond-norm analyses reveal different scaling behavior across gate sets and error models.
QPT WITH OVERSAMPLING
- Oversampling does not compensate for general SPAM errors, but a unitary 2-design removes some uniform SPAM errors.The distinction concerns increasing the number of independent experiments or gates, rather than merely repeating measurements.Source: QPT WITH OVERSAMPLING · p. 4
- Fig. 3 compares reconstructed-identity error against average SPAM error across four gate sets and three SPAM-error types using gate fidelity and diamond norm.Source: QPT WITH OVERSAMPLING · p. 4, Figure 3 · p. 5
- For gate-independent errors, the reconstruction error has polynomial scaling with the gate-set error because the errors effectively commute with reconstruction.Source: QPT WITH OVERSAMPLING · p. 5
- Unitary 2-design twirling converts the relevant factors into depolarizing channels, yielding a simplified bare estimate for constant gate-set errors.Source: QPT WITH OVERSAMPLING · pp. 5–6
- Global unitary errors behave like depolarizing errors for fidelity under unitary 2-designs, whereas independent per-gate unitaries remain qualitatively similar across gate sets.The global-unitary case represents global frame misalignment rather than control errors such as over-rotation or detuning.Source: QPT WITH OVERSAMPLING · p. 6
- Diamond-norm reconstruction error scales polynomially with gate-set diamond-norm error for all tested SPAM-error types and gate sets.The fidelity and diamond-norm metrics therefore produce substantially different views of the same reconstruction errors.Source: QPT WITH OVERSAMPLING · p. 6
SELF-CONSISTENT TOMOGRAPHY
The paper replaces standard QPT's SPAM-sensitive likelihood with a self-consistent reconstruction of an entire gate library, using a linearization around the ideal gates. In simulations with random unitary errors, this approach improves accuracy relative to standard QPT, though it incurs substantial computational and numerical overhead.
Motivation and approach
- Self-consistent tomography estimates the gate library and SPAM-related error maps together rather than treating preparation and measurement gates as known.The method assumes little about SPAM errors and reconstructs the library in a self-consistent manner.Source: SELF-CONSISTENT TOMOGRAPHY · p. 6
Optimization
- The reconstruction minimizes a least-square estimator subject to physical error-map constraints, formulated as a semidefinite program.The physical maps are completely positive and trace preserving, while the enlarged optimization can have non-unique solutions because of frame invariance.Source: SELF-CONSISTENT TOMOGRAPHY · pp. 6–7
Linearized estimation
- Linearizing the gate evolution about the ideal set reduces the estimator's order and is intended for benchmarking faulty components rather than probing unknown interactions.The approximation writes each estimated error map as the identity plus a small perturbation and expands to first order.Source: SELF-CONSISTENT TOMOGRAPHY · p. 6
Numerical comparison
- For random unitary errors on a six-gate cardinal-direction library, self-consistent tomography shows exponential accuracy improvement as gate error decreases relative to standard QPT.The comparison uses both fidelity and diamond-norm errors, averaged over the six gates, and also reports error-to-gate-error ratios.Source: SELF-CONSISTENT TOMOGRAPHY · p. 7, Figure 4 · p. 7
Likelihood comparison
- The likelihood comparison shows standard QPT's estimator remains flat as gate error decreases, whereas the linearized estimator has more favorable scaling despite not exactly minimizing the target quantity.The residual mismatch is attributed to the linear approximation.Source: Figure 5 · p. 7, SELF-CONSISTENT TOMOGRAPHY · p. 8
Limitations
- The method increases experiment count and reconstruction complexity, with numerical precision limiting larger simulations and frame maximization adding substantial computational cost.The authors state that the overhead is polynomial in system dimension but non-trivial because dimension grows exponentially with qubit number.Source: SELF-CONSISTENT TOMOGRAPHY · p. 8
EXPERIMENT
The method was applied to two independent single-qubit transmon experiments using the gate set {I, Xπ, Xπ/2, Yπ/2}. Self-consistent tomography produced lower estimated errors than QPT, while randomized benchmarking was lower still, especially for sample B.
Experimental systems
- The experiments used two independent single-junction transmon systems and the gate library {I, Xπ, Xπ/2, Yπ/2}.Sample A was capacitively coupled to a coplanar-waveguide resonator, whereas sample B was mounted in a copper cavity.Source: EXPERIMENT · p. 8
Experimental comparison
- QPT estimated roughly 1.5 times sample A’s and 3 times sample B’s error compared with self-consistent tomography across fidelity and diamond-norm measures.Table I compares both estimation methods with the desired gates for each sample.Source: EXPERIMENT · p. 8
Reconstruction visualization
- Figure 6 compares ideal, QPT-reconstructed, and self-consistent R-matrices for each of sample B’s four measured gates.The figure provides a gate-by-gate visual comparison of the two reconstructions against the ideal maps.Source: EXPERIMENT · p. 8
Benchmark comparison
- Randomized benchmarking produced smaller fidelity errors than self-consistent reconstruction, dramatically so for sample B.The authors conjecture that drifting calibration of the measurement operator M0 contributes because randomized benchmarking is insensitive to this error.Source: EXPERIMENT · p. 8
CONCLUSION
The paper concludes that standard QPT can fail when preparation and measurement components are faulty, and proposes self-consistent reconstruction of the full gate library instead. The method gives more trustworthy, generally higher-fidelity estimates in simulation and experiment, with the same experiments and polynomially greater classical processing.
Conclusion
- Standard process tomography can fail dramatically when all components of the experiment, including preparation and measurement, are faulty.The conclusion states that statistical analysis alone cannot improve reconstruction without incorporating these errors into the model.Source: CONCLUSION · p. 8
- The protocol modifies QPT’s likelihood function to incorporate uncertainty in state-preparation and measurement gates while reconstructing the entire gate library self-consistently.It uses the same number of experiments as QPT and adds only polynomial classical post-processing overhead.Source: CONCLUSION · p. 8, Table I · p. 9
- The full-information protocol is experimentally burdensome for systems larger than one or two qubits because its required experiment count can become excessive.The authors describe this burden as inherent to protocols that provide full information about a gate library.Source: CONCLUSION · p. 8, Table I · p. 9
- Self-consistent reconstruction yields more trustworthy and generally higher-fidelity gate estimates than tomography of individual gates in both simulation and experiment.The authors expect the difference from standard QPT to become more pronounced as gate errors decrease.Source: CONCLUSION · p. 9