Source-linked AI summary
Detector Tomography on IBM 5-qubit Quantum Computers and Mitigation of Imperfect Measurement
Yanzhu Chen, Maziar Farahzad, Shinjae Yoo, Tzu-Chieh Wei
TL;DR
IBM Q 5 readout measurements are imperfect, motivating characterization of the detector POVMs rather than assuming ideal projectors. The authors apply quantum detector tomography to Tenerife and Yorktown, find deviations from ideal projectors and unusual qubit-3 behavior, and describe using the characterized POVMs to estimate ideal detection statistics.
Problem
Readout errors on IBM Q devices are non-negligible, so treating measurements as ideal projectors can misrepresent detection statistics.
Source: §I · p. 1, §II · p. 2Method
The authors use quantum detector tomography to estimate measurement POVMs on IBM Q 5 Tenerife and Yorktown from prepared test states and measured frequencies.
Source: §II · p. 3, §III · p. 4Results
The characterized POVMs deviate from ideal projectors, and Tenerife qubit 3 behaves differently when measured alone versus with other qubits operating.
Source: §I · p. 2, §VI · p. 11Takeaways & Limitations
The characterized POVMs can support first-order correction and estimation of ideal measurement statistics despite their deviation from ideal projectors.
Source: §I · p. 2, §VI · p. 11A five-qubit detector model requires 7776 circuits using the stated basis states, and compressed sensing may mitigate this burden.
Source: §III · p. 4
Abstract
from arXiv · showhide
We use quantum detector tomography to characterize the qubit readout in terms of measurement POVMs on IBM Quantum Computers IBM Q 5 Tenerife and IBM Q 5 Yorktown. Our results suggest that the characterized detector model deviates from the ideal projectors by a few percent. Further improvement on this characterization can be made by adopting two- or more-qubit detector models instead of independent single-qubit detectors for all the qubits in one device. An unexpected behavior was seen in the physical qubit labelled as qubit 3 of IBM Q 5 Tenerife, which can be a consequence of detector crosstalk or qubit operations influencing each other and requires further investigation. This peculiar behavior is consistent with characterization from the more sophisticated approach of the gate set tomography. We also discuss how the characterized detectors' POVM, despite deviation from the ideal projectors, can be used to estimate the ideal detection distribution.
I. INTRODUCTION
NISQ devices require tools to characterize and mitigate errors, and detector tomography addresses readout by estimating measurement POVMs. The paper places this method alongside state and process tomography and describes an independent single-qubit model that can be extended to multiple qubits.
I. INTRODUCTION
- Existing state- and process-tomography methods can depend on accurate state preparation or measurement and do not scale favorably with system size.Randomized benchmarking can estimate average gate error independently of state-preparation and measurement errors, but provides partial characterization.Source: §I · p. 1
- IBM readout errors of 2–10% are comparable to two-qubit gate errors of 3–7% and exceed single-qubit gate errors of 0.1–0.2%.The authors use this disparity, together with relatively high-fidelity short-circuit state preparation, to motivate detector tomography.Source: §I · p. 1
II. TOMOGRAPHIC TOOLS FOR STATES, DETECTORS AND PROCESSES
- Quantum process tomography characterizes a process by applying it to a complete basis of density matrices and measuring the outputs.The paper also describes an alternative using a bipartite maximally entangled input and tomography of the resulting state.Source: §II · p. 2
- Detector tomography estimates unknown measurement operators from known test states, complementing state tomography’s estimation of unknown states from known measurement operators.The setup assumes test states are known or have smaller errors than detection; measurement in bases other than Z is made by applying rotations before Z measurement.Source: §II · p. 3
- The single-qubit model fits POVM parameters to frequencies from six Pauli-eigenstate preparations, assuming negligible inter-qubit crosstalk.The model requires POVM positivity and normalization; the authors note that it can be relaxed to a multi-qubit model with binary-string outcomes.Source: §II · p. 3
- Characterized detectors can support state tomography that mitigates detector errors, and some correction is possible without informationally complete state measurements.The paper also describes inferring ideal measurement statistics from characterized readout.Source: §I · p. 2, §II · p. 3
A. Maximum likelihood estimation
The detector POVM is estimated by maximizing a likelihood built from observed outcome frequencies for an informationally complete set of test states. Iterative updates enforce normalization, while the stopping threshold is chosen to keep numerical error below statistical uncertainty.
A. Maximum likelihood estimation
- The likelihood uses observed frequencies for outcomes on an informationally complete set of test states, with the POVM constrained to sum to identity.The frequency for outcome n on state ρ_i is compared with the probability predicted by POVM element Π(n).Source: A. Maximum likelihood estimation · p. 3
- Each iteration updates the POVM elements and then calculates the normalization-constraint term R for the next update.The term R depends on the POVM through the predicted outcome probabilities.Source: A. Maximum likelihood estimation · p. 3
- The iteration stops when successive POVMs differ by less than cutoff ϵ, which should keep numerical error below uncertainty from statistical fluctuations.Positivity and normalization are preserved when the initial POVM elements form a POVM.Source: A. Maximum likelihood estimation · p. 3
III. RESULTS OF QUANTUM DETECTOR TOMOGRAPHY
The experiments characterize readout on IBM Q 5 Tenerife and Yorktown using maximum-likelihood estimation and a single-qubit detector model. Individual and parallel measurements show discrepancies despite the model’s independence assumption, while a five-qubit model would require 7776 circuits.
III. RESULTS OF QUANTUM DETECTOR TOMOGRAPHY
- Individual and parallel measurements should differ only by statistical fluctuations under the independent single-qubit model, but the experiments show significant discrepancies.The study performs QDT on IBM Q 5 Tenerife and Yorktown, preparing test states with single-qubit gates and fitting measured frequencies by maximum likelihood.Source: §III · p. 4
- The authors identify a multi-qubit detector model as a next step, but report that two-qubit tomography does not capture the observed discrepancy solely as pairwise influence.A five-qubit model is described as the most complete characterization for these five-qubit devices.Source: §III · p. 4
- 7776 circuits are required to obtain all 32 operators in the five-qubit detector model using the stated basis states.The authors note that compressed sensing may reduce this burden and that the circuits must be divided into smaller jobs on current devices.Source: §III · p. 4
A. Single qubit detector: parallel vs. individual
Single-qubit detector estimates depart from ideal projectors, and individual and simultaneous measurements can yield distinct detector models. Pairwise comparisons indicate cross-qubit influences, while Tenerife qubit 3 shows behavior not captured by any one pairwise estimate.
A. Single qubit detector: parallel vs. individual
- Detector axes mostly align with z, but weights depart from 1/2 and transverse components are slightly nonzero.These departures distinguish the characterized POVMs from ideal 0/1 projectors.Source: A. Single qubit detector: parallel vs.… · p. 4
- Tenerife qubit 3 reverses the usual outcome-weight pattern under parallel readout; individual readout lacks this exception, hinting at influence from other qubits.For almost all qubits, the outcome-0 weight exceeds the outcome-1 weight, attributed to relaxation toward |0⟩.Source: A. Single qubit detector: parallel vs.… · p. 4
- Individual-versus-parallel detector-vector distances are an order of magnitude above O(10^-3) fluctuations, indicating correlations during simultaneous operation.Table I compares distances for the two devices.Source: A. Single qubit detector: parallel vs.… · p. 4, Table I · p. 4
B. Beyond single qubit detector
- Two-qubit QDT uses 64 parameters for four outcomes; singular values assess violations of the independent-detector assumption, though detailed analysis is omitted.The parameters are organized into four 4 × 4 matrices.Source: B. Beyond single qubit detector · pp. 4–5
- Tracing out one detector from a pair gives a conditioned single-qubit model, usually consistent within statistical fluctuations with pairwise parallel measurements, with exceptions in both devices.The reported exceptions are Tenerife qubit 3 conditioned on qubit 2 and Yorktown qubit 1 influenced by qubit 2.Source: B. Beyond single qubit detector · pp. 5–7
- Across both devices, distances exceed statistical resolution by at least an order of magnitude, suggesting pairwise influence and possible gains from two-qubit models.The comparisons are presented for Tenerife and Yorktown in Tables III and IV.Source: B. Beyond single qubit detector · p. 7, Table III · p. 7, Table IV · p. 8
- Tenerife qubit 3’s all-qubit parallel result differs from every pair-conditioned result, implying nontrivial correlation when several physical qubits operate together.The pair-conditioned comparisons trace out each other qubit from two-qubit detectors.Source: B. Beyond single qubit detector · p. 7, Table III · p. 7
- A three-qubit model was demonstrated on connected qubit triplets using 50 runs of 8192 shots, with data used for first-step correction.The model has 512 parameters organized across eight outcome matrices.Source: B. Beyond single qubit detector · pp. 7–8
C. Error analysis
The uncertainty analysis uses non-parametric bootstrap resampling of repeated experimental runs, with the standard deviation across resampled estimates representing statistical fluctuation.
C. Error analysis
- Uncertainty was estimated by resampling experimental data sets with replacement and calculating the standard deviation across the resulting parameter estimates.The authors identify this standard deviation as an estimate of uncertainty due to statistical fluctuation.Source: C. Error analysis · p. 8
- Each experiment was repeated 100 times with 8192 shots per run, yielding 819200 effective shots for the MLE estimate.Source: C. Error analysis · p. 8
- The analysis used 100 resampled data sets; selected checks with up to 1000 resamples produced similar standard deviations.Source: C. Error analysis · p. 8
IV. APPLICATION OF CHARACTERIZED DETECTORS—INFERRING IDEAL DETECTION
The section describes inferring ideal computational-basis outcome probabilities from measured frequencies using characterized detectors. The procedure relies on stated assumptions and offers constrained optimization when direct inversion yields negative probabilities.
IV. APPLICATION OF CHARACTERIZED DETECTORS—INFERRING IDEAL DETECTION
- Under the assumption of no detector crosstalk, characterized detectors relate measured frequencies to an ideal joint outcome distribution that can be inferred from existing data.The notation distinguishes experimental distribution ˜P from ideal distribution P.Source: §IV · p. 8
- When |a_1| and |a_2| are much smaller than |a_3|, the measurement relation simplifies to a left-stochastic matrix M that can be inverted to obtain ideal probabilities.The matrix elements are described over all possible outcomes.Source: §IV · pp. 8–9
- If direct inversion produces negative probabilities, minimize |MP − ˜P|^2 subject to positivity and normalization instead.The authors describe this as a convex quadratic programming problem.Source: §IV · p. 9
- This first-step correction does not match the full power of QST with characterized detectors, but requires only computational-basis frequencies and no further experiments.Source: §IV · p. 9
- When coefficients involving Pauli indices 1 or 2 are not negligible, the averaging procedure can still be used by running additional circuits with appended Pauli Z gates.The procedure's stated conclusion is based on the detector-POVM assumption in p42.Source: §IV · pp. 9–10
- Averaging probabilities across circuits with appended Pauli Z gates cancels terms involving a_1 and a_2 and yields the stated average probability.The circuits span 2^N choices of the binary string K, including the original circuit.Source: §IV · p. 9
A. Using characterized detectors
The authors apply first-step correction to a Yorktown circuit and to Bell- and GHZ-state measurements. The corrected distributions move the dominant outcomes toward their ideal values.
A. Using characterized detectors
- In IBM Q 5 Yorktown, Hadamard on qubit 3 followed by CNOT from qubit 3 to qubit 4 and measurement ideally yields P00000 = P11000 = 0.5.The circuit ran 50 times with 8192 shots per run; the two largest measured components were P̃00000 = 0.466 and P̃11000 = 0.422.Source: A. Using characterized detectors · p. 10
- For the Yorktown circuit, correction yielded P_00000 = 0.493 and P_11000 = 0.507 with other components of order O(10^-17); parallel calibration yielded 0.495 and 0.505.The ideal distribution assigns probability 0.5 to each of these two outcomes.Source: A. Using characterized detectors · p. 10
- For Bell- and GHZ-state measurements, first-step correction brought P_00 and P_11, or P_000 and P_111, closer to 0.5 and reduced their difference.Direct inversion with a zero cutoff and SLSQP optimization gave similar results.Source: A. Using characterized detectors · p. 10
V. GATE SET TOMOGRAPHY
The section outlines gate set tomography (GST), which estimates a gate set comprising an initial state, gates, and a two-outcome POVM. Its analysis addresses a gauge freedom before a final maximum-likelihood estimate.
V. GATE SET TOMOGRAPHY
- GST models an unknown initial state, a set of unknown CPTP gates, and an unknown two-outcome POVM as a gate set.Source: §V · p. 10
- Linear GST expresses the gate set in an arbitrary basis using experimentally obtained inner products and informationally complete gates for the state and measurement.Source: §V · p. 10
- The resulting gate set is identifiable only up to an invertible transformation, so gauge optimization matches it toward a target; long gate sequences amplify gate-parameter errors, but not SPAM errors.Source: §V · p. 10
- The full GST procedure proceeds through linear GST, gauge optimization, iterative data addition for χ2 minimization, and final MLE analysis with gauge optimization.Source: §V · p. 10
A. Results of single-qubit detector characterization from GST
GST detector estimates for the five-qubit devices agree within a few percent with simple detector tomography, while qubit 3 of Tenerife behaves differently in parallel and individual measurements.
A. Results of single-qubit detector characterization from GST
- The GST analysis used an initial LGST estimate, gauge optimization, and successive long-sequence maximum-likelihood fitting; nonpositive POVMs triggered a SPAM-penalty adjustment.The reported setup used 272 circuits, compared with six for QDT, and 8192 shots per circuit.Source: A. Results of single-qubit detector… · p. 11
- GST detector POVMs on the two five-qubit devices agree within a few percent with the earlier simple detector tomography results.The reported GST circuits were run in parallel across all five qubits on both devices.Source: A. Results of single-qubit detector… · p. 11
- In parallel measurement, Tenerife qubit 3 has a( n=0 )_0 = 0.4521, smaller than a( n=1 )_0 = 0.5479.The reported comparison describes the unusual parallel-measurement behavior.Source: A. Results of single-qubit detector… · p. 11
- The qubit-3 GST detector estimate from individual measurement is closer to simple detector tomography, with about 4% difference.The individual GST parameters are reported as (0.5182, 0.0036, 0.0022, 0.4449) and (0.4818, −0.0036, −0.0022, −0.4449).Source: A. Results of single-qubit detector… · p. 11
VI. CONCLUSION AND DISCUSSION
The paper applies standard detector tomography to IBM Q 5 Tenerife and Yorktown under assumptions about state preparation and single-qubit gates. The resulting POVMs deviate from ideal projectors, and the characterization indicates a qubit-3 anomaly on Tenerife.
VI. CONCLUSION AND DISCUSSION
- The detector POVMs deviate from ideal projectors and can be used for first-order correction in experiments.The characterization assumes negligible errors in ground-state preparation and in the single-qubit gates used to prepare Pauli eigenstates.Source: §VI · p. 11
- The paper reports evidence of qubit crosstalk on one device, with Tenerife qubit 3 showing different behavior in individual and parallel measurement.The GST characterization also reports distinct parallel and individual qubit-3 estimates.Source: §VI · p. 11, Table VI · p. 12
Appendix A: Tables
The appendices report detector-parameter uncertainty and visualizations for Yorktown, plus a parallel-measurement characterization of all 14 qubits on Melbourne.
Appendix A: Tables
- The estimated error for each processed experimental parameter is typically O(10^-4), with the largest reported error up to 0.003.The Yorktown detector parameters are also plotted across 100 runs.Source: Appendix A · p. 13
Appendix B: QDT for IBM Q 16 Melbourne
- On Melbourne, all 14 detectors align mostly along the vertical z-axis, while qubit 3 has comparatively shorter arrows and the largest imperfection.The reported characterization used parallel measurements across all 14 qubits; individual, pairwise, and triple characterizations are not presented.Source: Appendix B · p. 13
Appendix A: Tables
- Figure 3 displays Yorktown single-qubit detector parameters for both individual and parallel measurements, including both POVM outcomes.The plotted components are ar_x,y,z = a_1,2,3.Source: Figure 3 · p. 14
Appendix C: Relaxation from GST
The appendix uses GST identity-gate estimates to examine relaxation after excitation. The inferred T1 depends on whether the gate is constrained to be trace preserving, and the CPTP-projected matrix does not yield a meaningful relaxation time.
Appendix C: Relaxation from GST
- The unconstrained GST identity gate gives T1 ≈29.5 µs, from exponential decay of the probability of measuring |1⟩ after excitation and repeated idling.The sequence applies an ideal σx to |0⟩, then m discrete identity gates; each single-qubit gate duration, including buffer time, is 70 ns.Source: Appendix C · pp. 14–15
- The unconstrained identity-gate estimate is not positive, so simulating other processes with it can produce negative, unphysical probabilities.The text gives T2 decoherence as an example of such a simulation.Source: Appendix C · p. 15
- Projecting the identity gate to be TP gives T1 ≈43.2 µs, about 50% different from the unconstrained estimate.The TP-projected result is shown in Fig. 5b.Source: Appendix C · p. 15
- The CPTP-projected identity-gate matrix does not provide a meaningful relaxation time, and a correct CPTP description of idle-gate relaxation or dephasing remains outside the paper’s scope.Figure 5 compares estimates without TP or CPTP constraints and after TP and CPTP projection.Source: Appendix C · pp. 15–16, Figure 5 · p. 16