PaperlayerView exact revision on arXiv

Source-linked AI summary

Experimental quantum compressed sensing for a seven-qubit system

C. A. Riofrio, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, J. Eisert

arXiv:1608.02263v1quant-ph

TL;DR

Quantum tomography becomes difficult as system size grows, and incomplete noisy data may not support reliable estimates of every density-matrix parameter. This work implements compressed-sensing tomography for a seven-qubit experiment and proposes support identification based on resampled estimates. The authors report faithful reconstructions and argue that low-rank estimates can remain appropriate when only leading eigenvectors are reliably recovered.

Abstract

from arXiv · show

Well-controlled quantum devices with their increasing system size face a new roadblock hindering further development of quantum technologies: The effort of quantum tomography---the characterization of processes and states within a quantum device---scales unfavorably to the point that state-of-the-art systems can no longer be treated. Quantum compressed sensing mitigates this problem by reconstructing the state from an incomplete set of observables. In this work, we present an experimental implementation of compressed tomography of a seven qubit system---the largest-scale realization to date---and we introduce new numerical methods in order to scale the reconstruction to this dimension. Originally, compressed sensing has been advocated for density matrices with few non-zero eigenvalues. Here, we argue that the low-rank estimates provided by compressed sensing can be appropriate even in the general case. The reason is that statistical noise often allows only for the leading eigenvectors to be reliably reconstructed: We find that the remaining eigenvectors behave in a way consistent with a random matrix model that carries no information about the true state. We report a reconstruction of quantum states from a topological color code of seven qubits, prepared in a trapped ion architecture, based on tomographically incomplete data involving 127 Pauli basis measurement settings only, repeated 100 times each.

Physical system and raw data

The experiment prepared seven trapped-ion qubits in a topological color-code state and characterized it using incomplete Pauli-basis measurements. The selected settings yield outcome statistics that are noisy because each setting is measured a limited number of times.

Physical system and prepared states

  • The trapped-ion platform uses 40Ca+ ions in a linear Paul trap, with qubits encoded in S1/2 and metastable D5/2 states and manipulated by laser pulses.The gate operations include collective and entangling operations as well as localized single-qubit phase rotations.Source: Physical system and raw data · p. 2
  • The seven physical qubits were prepared in a topological color-code state whose code space is spanned by the logical states |¯0⟩ and |¯1⟩.The anticipated states are those the ideal experiment would prepare; statistical noise and systematic errors limit preparation knowledge.Source: Physical system and raw data · p. 2

Measurement settings

  • The experiment selected 127 Pauli-basis settings, compared with 2187 settings for an informationally complete set.Each setting selects a local Pauli matrix for each qubit, and each qubit has two possible measurement outcomes.Source: Physical system and raw data · pp. 2–3

Measurement statistics

  • Each setting was repeated 100 times, and relative outcome frequencies were used to estimate measurement probabilities.With 2^L possible outcomes per setting and relatively few repetitions, many possible outcomes may not appear.Source: Physical system and raw data · p. 3

Measurement settings

  • Pauli-basis measurements provide 2^L outcome probabilities per local-Pauli setting, whereas Pauli-correlation measurements provide one expectation value per setting.The basis measurements arise naturally in the ion-trap architecture, and Pauli correlations can be recovered from them.Source: Physical system and raw data · p. 3

Estimators and state reconstruction

The section compares trace-norm minimization, positivity-constrained least squares, and spectral thresholding for reconstructing the prepared states. The reported reconstructions are valid and faithful, but estimator choice changes fidelity and how the spectral tail is represented.

Estimators and state reconstruction

  • The matrix Lasso adds a nuclear-norm penalty to encourage low-rank solutions, while positivity-constrained least squares uses μ = 0 with a positive-semidefinite constraint.The trace norm is described as the tightest convex relaxation of rank.Source: Estimators and state reconstruction · p. 3
  • GRAD optimizes a parametrization that is positive semidefinite by construction, avoiding the costly projection step needed to enforce positivity during optimization over ρ.The simplification improves computational effort relative to earlier iterative gradient methods based directly on ρ.Source: Estimators and state reconstruction · p. 3
  • The estimators produced valid reconstructions close in fidelity to the anticipated states, with runtimes of a few hours or a few minutes for GRAD.The authors present this as a first experimental implementation of compressed-sensing tomography for high-dimensional quantum systems.Source: Estimators and state reconstruction · p. 4
  • Fidelity ranged from F = 0.98 for trace norm minimization to F = 0.30 for least squares on the same data.The authors hypothesize that limited data and the trace-norm penalty against mixed states contribute to the difference.Source: Estimators and state reconstruction · p. 4
  • Trace-norm estimates emphasize a few leading eigenvalues, while least-squares and spectral-thresholding estimates retain heavier, exponentially decaying tails.The authors suggest that leading-eigenvector estimates may help diagnose coherent errors, whereas estimates retaining the tail may help diagnose incoherent noise.Source: Estimators and state reconstruction · p. 4

Quantum support identification

With incomplete, noisy data, a full density-matrix estimate may be neither reliable nor actionable. The section proposes identifying support by retaining eigencomponents that are reproducible across resampled estimates and exceed a random-overlap threshold.

Quantum support identification

  • Limited data may leave a full density matrix unreliable and overfit statistical fluctuations, while leading eigenvectors can contain more actionable information about dominant errors.The paper notes that the practical value of knowing very low-ranked eigenvectors is unclear.Source: Quantum support identification · pp. 4–5
  • In the stabilizer-basis reconstruction, the trace-norm estimate concentrates support in a few diagonal elements, while least-squares and spectral-thresholding estimates have heavier tails.The plotted diagonal elements decay exponentially after an initial rapid drop.Source: Figure 2 · p. 5
  • The proposed heuristic discards estimated parameters consistent with random-matrix noise, extending to more general settings without the rigorous guarantees of an earlier method.The earlier spectral-thresholding method was analyzed for informationally complete measurements.Source: Quantum support identification · p. 5
  • Only some estimated eigenprojections are stable across data realizations; unsupported directions fluctuate with the particular data set.The protocol compares eigenprojections across estimates to identify components supported by the data.Source: Quantum support identification · p. 5
  • Bootstrap resampling generates multiple density-matrix estimates, and the protocol selects the largest rank whose eigenvector overlap exceeds the random-overlap threshold.The threshold is based on the expected overlap and variance for Haar-random state vectors.Source: Quantum support identification · pp. 5–6

Conclusion and perspectives

The paper frames tomography as a way to learn how an experimental state deviates from its target and guide apparatus changes. It presents compressed tomography and argues that only statistically supported low-rank states should be reported.

Conclusion and perspectives

  • Tomography can guide apparatus changes by revealing how the prepared state deviates from the anticipated state.The stated aim goes beyond certifying whether a preparation matches its target.Source: Conclusion and perspectives · p. 6
  • The work advocates reporting only low-rank states with a statistical basis when tomography uses informationally incomplete data.It frames this as a question of what state tomography can mean for intermediate-sized systems.Source: Conclusion and perspectives · p. 6
  • The paper presents a successful first compressed-sensing tomography implementation on a moderately sized quantum experiment.It also discusses quantum support identification for reconstruction from incomplete data.Source: Conclusion and perspectives · p. 6

Grad estimator

The GRAD estimator optimizes a factor Q of the state rather than the state matrix directly, so positivity is built into the parameterization. This avoids the projection step used by earlier gradient approaches.

Grad estimator

  • Parameterizing the density matrix to make it manifestly positive semidefinite lets GRAD optimize over Q rather than ρ.The estimator uses a gradient search on Q with the vector 2-norm objective.Source: Grad estimator · p. 7
  • The Q-based gradient can be computed analytically, and the iteration continues until convergence before the state is renormalized.The trace is not constrained during optimization.Source: Grad estimator · pp. 7–8
  • Earlier gradient methods update ρ and use a projector to enforce positivity, whereas GRAD incorporates positivity directly through its parameterization.The projection-based approach works but significantly slows the algorithm.Source: Grad estimator · pp. 7–8

Random matrices

The supplementary analysis considers overlaps of random vectors in C^d under the Haar measure. It computes their first and second moments using Weingarten functions and derives a variance expression.

Random matrices

  • The analysis considers a random variable for an arbitrary vector in C^d and its moments under the Haar measure.The quantity is identified as the overlap of two random vectors.Source: Random matrices · p. 8
  • The first and second moments of the random-vector overlap are computed using the Weingarten function formalism.Source: Random matrices · p. 8
  • The second-moment calculation yields an expression for the variance of the overlap quantity.The supplied passage does not state the expression's value.Source: Random matrices · p. 8

Further estimators

The section reviews positivity-constrained least squares and trace-norm minimization as estimators for incomplete quantum tomography. Their practical feasibility depends on system size and measurement count, and TNM also requires choosing an error level.

Further estimators

  • LS-SDP is a least-squares semidefinite program with a positivity constraint on the estimated state.The positivity constraint restricts estimates to the positive cone while retaining compatibility with the data.Source: Further estimators · p. 8
  • For L = 7 and n = 127 measurement settings, LS-SDP remains usable, and it can estimate well from incomplete data when the state is not too mixed.Its matrix-form computation scales unfavorably in effort and memory with system size.Source: Further estimators · p. 8
  • TNM minimizes trace norm with a positivity constraint, resembles the Dantzig selector, and remains feasible for seven qubits with fewer than about a thousand settings on a standard workstation.The error level ε must be estimated, and the estimator is designed to produce low-rank estimates.Source: Further estimators · pp. 8–9

Simulations on spectral thresholding

Simulations test whether the rank-selection protocol identifies true state rank and how it compares with spectral thresholding. In the low-data regime, it tends to select lower ranks, while the comparison method estimates higher ranks and shows similar risk on average.

Simulations on spectral thresholding

  • With enough measurements, the protocol can distinguish the true rank, but in the low-data regime it tends to choose a lower rank, avoiding an unsupported higher-rank fit.Simulations used four-qubit states of ranks 1, 2, 4, and 8, varying measurement settings and repetitions.Source: Simulations on spectral thresholding · p. 9
  • Risk is assessed by comparing the true state with the spectrally thresholded reconstruction; the risk curves are shown in fig. 9.The comparison between methods reports no clear risk trend beyond similar average risk.Source: Simulations on spectral thresholding · p. 9
  • The comparison adapts ref. 43’s thresholding algorithm, originally using unconstrained least squares for informationally complete measurements, to positivity-constrained least squares and incomplete data.Its threshold parameter follows ref. 43; the reconstructed spectrum is renormalized by shifting eigenvalues after thresholding.Source: Simulations on spectral thresholding · p. 9
  • With positivity imposed, ref. 43’s spectral-thresholding method estimates higher ranks than this protocol with insufficient measurements, while their average risks appear similar.The comparison applies the method naively to informationally incomplete measurements; rigorous guarantees for spectral thresholding in that regime were unavailable.Source: Simulations on spectral thresholding · p. 9

Additional results

Two additional laboratory-prepared states are reconstructed: the logical |¯1⟩ state and the encoded superposition. Their results show the estimator contrast seen earlier, with spectral thresholding also applied successfully.

Additional results

  • For two additional prepared states, trace-norm minimization delivers an almost-pure reconstruction while positivity-constrained least squares does not; spectral thresholding also succeeds.The comparison is reflected in fidelities to the anticipated states.Source: Additional results · p. 9
  • Figures 3–6 plot mean eigenvector overlap against rank truncation for true states of ranks 1, 2, 4, and 8, with the eq. (16) threshold marked.The captions specify the plotted quantities and threshold, not the numerical outcomes.Source: Figure 3 · p. 10, Figure 4 · p. 10, Figure 5 · p. 10
  • Figure 7 plots the estimated rank against measurement settings and repetitions, defaulting to rank one when the criterion cannot return a nonzero rank.Source: Figure 7 · p. 11
  • Figure 8 gives mean rank from positivity-constrained spectral thresholding; its caption notes some low-measurement overfitting and that the mean need not be discrete.Source: Figure 8 · p. 11
  • Figures 9 and 10 show mean risk for the paper’s thresholding procedure and the positivity-constrained ref. 43 procedure, respectively.The captions identify the risk plots and methods but do not provide unseen values.Source: Figure 9 · p. 12, Figure 10 · p. 12
  • Figures 11 and 12 display absolute density-matrix entries for reconstructions of the logical |¯1⟩ state and the encoded superposition, respectively.Source: Additional results · p. 9, Figure 11 · p. 12, Figure 12 · p. 13
Loading 1608.02263v1…