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Time Series Quantum Reservoir Computing with Weak and Projective Measurements

Pere Mujal, Rodrigo Martínez-Peña, Gian Luca Giorgi, Miguel C. Soriano, Roberta Zambrini

arXiv:2205.06809v1quant-phcond-mat.dis-nn

TL;DR

Quantum reservoir computing faces the challenge of extracting time-series information efficiently without losing reservoir memory to measurement back-action. The paper analyzes repetition-based and weak-measurement protocols, finding that suitable choices can reach optimum performance beyond ideal assumptions in memory and forecasting tasks. The appropriate protocol depends on fading memory and task requirements.

  • Problem

    Realistic QRC time-series processing must address measurement resources and back-action while retaining reservoir memory.

    Source: §I · pp. 1–2, A. Restarting, rewinding and online… · p. 3

  • Method

    The paper analyzes restarting and rewinding protocols with projective measurements and an online protocol using weak monitoring.

    Source: §III · p. 8

  • Results

    Optimum performance is reachable beyond ideal assumptions with projective RSP and RWP or weakly monitored OLP, and suitable OLP measurement strength can exceed RWP performance depending on fading memory and task.

    Source: §III · pp. 8–9

  • Takeaways & Limitations

    OLP operates without storing input data, while protocol efficiency depends on fading memory and the task.

    Source: §III · pp. 8–9

Abstract

from arXiv · show

Quantum machine learning represents a promising avenue for data processing, also for purposes of sequential temporal data analysis, as recently proposed in quantum reservoir computing (QRC). The possibility to operate on several platforms and noise intermediate-scale quantum devices makes QRC a timely topic. A challenge that has not been addressed yet, however, is how to efficiently include quantum measurement in realistic protocols, while retaining the reservoir memory needed for sequential time series processing and preserving the quantum advantage offered by large Hilbert spaces. In this work, we propose different measurement protocols and assess their efficiency in terms of resources, through theoretical predictions and numerical analysis. We show that it is possible to exploit the quantumness of the reservoir and to obtain ideal performance both for memory and forecasting tasks with two successful measurement protocols. One consists in rewinding part of the dynamics determined by the fading memory of the reservoir and storing the corresponding data of the input sequence, while the other employs weak measurements operating online at the trade-off where information can be extracted accurately and without hindering the needed memory. Our work establishes the conditions for efficient protocols, being the fading memory time a key factor, and demonstrates the possibility of performing genuine online time-series processing with quantum systems.

I. INTRODUCTION

Quantum reservoir computing (QRC) brings reservoir computing to quantum systems, but realistic time-series processing must account for both finite measurement resources and measurement back-action. This work compares measurement protocols to identify strategies balancing accuracy, resource demands, and reservoir memory.

I. INTRODUCTION

  • Quantum measurements require ensembles of system copies and generally alter the measured state, creating linked challenges for temporal processing.The paper frames accuracy and back-action as concerns for continuous monitoring across sequential inputs.Source: §I · pp. 1–2
  • QRC has been proposed for temporal tasks across quantum models and platforms, with potential relevance to noisy intermediate-scale quantum devices.Prior work includes a quantum digital computer exploration of temporal processing, while the paper identifies efficient measurement as an unresolved issue.Source: §I · pp. 1–2
  • The study assesses restarting, rewinding, and weak-measurement protocols using theoretical and numerical analysis of accuracy and resource requirements.It examines memory and forecasting tasks and seeks strategies that balance performance with resource efficiency.Source: §I · p. 2

A. Restarting, rewinding and online protocols

The section compares three ways to extract outputs while accounting for measurement back-action: restart the sequence, rewind only its fading-memory window, or monitor the reservoir online with weak measurements. Repetition-based protocols avoid incorporating measurement back-action into the dynamics but require repeated processing and external storage of inputs.

A. Restarting, rewinding and online protocols

  • RSP reinitializes and repeats the sequence for each output, requiring Nmeas experiment repetitions at each input time step.This avoids accounting for measurement back-action in the evolving dynamics, but incurs repeated processing.Source: A. Restarting, rewinding and online… · p. 3
  • RWP limits repetition to the washout interval τwo, using the reservoir’s fading memory to reduce the repeated input history.The washout time is the time for the dynamics to lose information about the initial state; the input history must still be stored for replay.Source: A. Restarting, rewinding and online… · p. 3
  • RSP and RWP estimate observables from finite measurement ensembles, while their repeated processing requires storing the corresponding input data externally.Finite ensembles introduce statistical error; repetition may cover all or only part of the input sequence.Source: A. Restarting, rewinding and online… · p. 3
  • OLP continuously monitors the evolving reservoir with tunable weak measurements, avoiding any repeated portion of its dynamics.Weak measurements perturb the state less than projective measurements and thereby help preserve time information.Source: A. Restarting, rewinding and online… · p. 3

B. Monitored qubit system

The monitored-system framework introduces a measurement superoperator into the reservoir’s discrete-time evolution and models indirect qubit measurements with tunable strength. The analysis relates measurement sharpness to decoherence and the size of the state disturbance.

B. Monitored qubit system

  • The OLP inserts a measurement superoperator into each discrete-time step to account for measurement effects during reservoir evolution.The monitored state evolves along quantum trajectories characterized by stochastic measurement outcomes.Source: B. Monitored qubit system · p. 3
  • Indirect qubit measurements use tunable strength g and outcome V; larger g approaches projective measurement, while smaller g makes outcomes less informative.For a single qubit measured along z, the mean outcome is g⟨σz⟩.Source: B. Monitored qubit system · p. 4
  • The measurement-induced state disturbance grows with sharper measurements, whereas very small g weakly perturbs the state.In the model, measurement introduces no nonlinearities beyond polynomials in input and measurement-outcome variables at different time steps.Source: B. Monitored qubit system · p. 4
  • The example QRC uses a qubit network evolving under a disordered transverse-field Ising Hamiltonian, with input injected by rewriting a node state.The selected parameters place the reservoir in an ergodic dynamical phase associated with optimal QRC performance.Source: B. Monitored qubit system · p. 4

C. Measurement accuracy

Measurement accuracy is constrained by finite ensemble size and, for continuous monitoring, by the weakness needed to limit back-action. The section compares these errors with the disturbance to reservoir dynamics, showing why measurement strength and observable choice matter.

C. Measurement accuracy

  • Sharp measurements reduce statistical uncertainty at fixed Nmeas, but OLP requires balancing that gain against back-action that can erase information.For weak measurements, matching single-qubit uncertainty requires Nmeas ∼ g^-2, versus Nmeas ∼ g^-4 for two-qubit observables.Source: C. Measurement accuracy · pp. 4–5
  • Finite-ensemble fluctuations can be small in plotted observables yet still affect machine-learning performance.Figure 2 compares OLP observables with unperturbed values across z and x directions and weak and strong measurement strengths.Source: C. Measurement accuracy · p. 5, Figure 2 · p. 5
  • Weak x-direction measurements approximately preserve unperturbed dynamics, whereas sharp x-direction measurements produce strong deviations through decoherence.The effect is less apparent for z measurements in this reservoir because its qubits are mostly aligned along z.Source: C. Measurement accuracy · p. 5
  • Weak measurement preserves the ideal state more closely but has greater finite-ensemble error than the unperturbed state.With an infinite ensemble, the estimation error for weak measurements vanishes.Source: C. Measurement accuracy · p. 5

D. Short-term memory capacity

The short-term memory task tests how well the reservoir reproduces past inputs under different measurement protocols. Weak online measurements preserve memory close to the unperturbed case, while sharp measurements hinder it; finite-shot performance depends on balancing measurement accuracy against back-action and resource use.

D. Short-term memory capacity

  • The task measures capacity to reproduce random past inputs at delay τ, with perfect short-term memory corresponding to C(τ) = 1.Source: D. Short-term memory capacity · p. 5
  • Weak online measurements at g = 0.3 approach unperturbed short-term memory, whereas sharp measurements hinder it; the reservoir stores information for about six previous steps.With ideal accuracy, sharp online measurements still fail to match the other protocols because back-action erases information in quantum coherences and correlations.Source: Figure 3 · p. 6, D. Short-term memory capacity · p. 6
  • Two-qubit correlations provide greater ideal memory capacity than single-qubit observables, but finite-measurement performance is worse because the correlations are smaller and noisier.The finite-count comparison is reported for the measured capacities; increasing Nmeas leaves room for improvement.Source: D. Short-term memory capacity · pp. 6–7
  • At finite measurement counts, one-qubit sum capacity is optimal around g = 0.3, although its ideal maximum occurs as g → 0+ when statistical uncertainty diverges.The optimal measurement becomes weaker as Nmeas increases.Source: D. Short-term memory capacity · p. 6, Figure 4 · p. 7
  • Around 10^6 measurements, RSP, RWP, and weak OLP approach their ideal capacities, while reset protocols perform better around 10^4 and strong OLP converges slowly to a lower value.The strong-measurement OLP appears to saturate below its infinite-measurement limit.Source: D. Short-term memory capacity · p. 7
  • For single-qubit observables, online processing can use fewer resources than rewinding when measurement strength meets a washout-step condition and back-action remains negligible.For Nwo = 20, g ≳ 0.23 is necessary; g = 0.3 gives negligible back-action, while the two-qubit condition is more restrictive.Source: D. Short-term memory capacity · p. 7

E. Chaotic time series forecasting

The paper evaluates forecasting on the chaotic Santa Fe laser time series. The reservoir achieves high one-step accuracy, while prediction quality generally declines with forecast distance; weak online measurements preserve performance closer to the unperturbed system than sharp measurements.

E. Chaotic time series forecasting

  • For one-step Santa Fe forecasts, combining single- and two-qubit observables reaches capacity C = 0.98, with similar accuracy from RSP or RWP.The benchmark is a chaotic NH3 laser time series.Source: E. Chaotic time series forecasting · pp. 7–8, Figure 5 · p. 8
  • Sharp online measurements at g = 10 produce lower prediction capacity than weak measurements at g = 0.3; in the infinite-measurement limit, weak-measurement capacity matches the unperturbed system.The simulated weak-measurement results could approach the upper bound with more measurements.Source: Figure 5 · p. 8, E. Chaotic time series forecasting · p. 8
  • For future predictions at distance η ≥ 1, goodness decreases as η increases; the reported local maxima in capacity are attributed to quasi-periodicity, with no locations specified.Source: E. Chaotic time series forecasting · p. 8

III. CONCLUSIONS

The conclusions identify restarting, rewinding, and online weak-measurement protocols for quantum reservoir computing, with performance and resource trade-offs shaped by measurement back-action and reservoir fading memory. The appendices derive ideal-case equations with back-action and extend the measurement treatment to the x and y directions.

III. CONCLUSIONS

  • The restarting and rewinding protocols use projective measurements and buffered inputs, while the online protocol uses weak monitoring without storing inputs externally.The conclusions report optimum performance beyond ideal assumptions for both projective-measurement protocols and online weak monitoring.Source: §III · p. 8
  • Online weak measurements partially preserve quantum coherences, and the conclusions identify superconducting qubits and trapped ions as implementation platforms.Measurement strength can be tuned through system–apparatus interaction time or photonic reflectivity.Source: §III · pp. 8–9
  • Depending on fading memory and task, online measurements at an appropriate strength could outperform rewinding and remain efficient for higher-order moments.Higher-order moments can be small and require more resources to estimate accurately.Source: §III · p. 9

Appendix A: Equations of the ideal case with back-action

  • The ideal-case derivation notes that same-direction measurements do not add a back-action contribution to the current observable expectation, though the updated state affects later time steps.The derivation also expresses the measurement matrix as a tensor product of single-qubit matrices.Source: Appendix A · p. 10

a. Measurements in the x and y directions

  • The appendix gives analogous measurement expressions for the x and y directions, with real and imaginary parts used in their probability distributions.For x-direction measurement, the state is rotated using Hadamard gates before measurement and rotated back afterward.Source: Appendix A · p. 10, a. Measurements in the x and y… · p. 11

b. Expectation values and uncertainties

The appendix characterizes statistical uncertainty for indirect measurements and describes the reservoir model and evaluation procedure. Weaker measurements generally require more samples to match the uncertainty of stronger measurements, while the uncertainty bounds apply across states and measurement directions.

b. Expectation values and uncertainties

  • For single-qubit measurements, the state- and direction-independent uncertainty bound is reached when the measured expectation value is zero.When measurement strength tends to zero or the expectation value tends to zero, state and direction dependence are small first-order corrections to the upper bound.Source: b. Expectation values and uncertainties · p. 12
  • For g < g′, more measurements are generally needed at strength g to match the statistical uncertainty at strength g′.The appendix also compares weak measurements with projective measurements.Source: b. Expectation values and uncertainties · p. 12

a. Expectation values and uncertainties

Appendix C: Quantum reservoir based on a disordered transverse Ising model

  • The simulated reservoir is a six-qubit disordered transverse-field Ising system, with real inputs encoded in one qubit at each time step.The external field and time interval are fixed at h = 10J_s and Δt = 10/J_s.Source: Appendix C · p. 13

1. QRC training and test

  • QRC predictions are formed as a linear weighted combination of reservoir observables, with weights fitted by least squares against training targets.The capacity metric ranges from 0 to 1, with 1 denoting perfect predictions.Source: §1 · p. 13
  • The STM and Santa Fe evaluations use separate train/test splits after discarding 20 washout steps.STM uses 1,000 time steps; Santa Fe uses 2,000 time steps.Source: §1 · p. 13

Appendix D: QRC performance considering different random qubit networks

The appendix checks whether the reported online-protocol behavior persists across random qubit-network realizations. It also describes the Gaussian-noise estimate used to represent finite-ensemble measurement fluctuations.

Appendix D: QRC performance considering different random qubit networks

  • No large capacity variations were observed across 100 random coupling realizations, and the qualitative behavior matched the single-network result.The comparison concerns STM sum capacity in the online protocol as measurement strength g varies.Source: Appendix D · p. 13, Figure 6 · p. 14
  • The noisy curves add a single zero-mean Gaussian fluctuation with standard deviation determined by the measurement uncertainty equations.The appendix describes this as an estimate of statistical noise from a limited ensemble size.Source: Appendix D · pp. 13–14
  • Figure 6 plots STM sum capacity against g for different observables, with ideal and Gaussian-noise estimates, coupling variation, and reference lines at g = 0.3 and g = 0.5.The plotted data are from the test set.Source: Figure 6 · p. 14

Appendix E: QRC performance using second order moments

Two-qubit observables preserve more memory at long delays than single-qubit observables in the unperturbed case, but online measurement strength and finite-sample noise limit that advantage. Rewind protocols approach unperturbed performance, while their measurement and experimental-time costs differ.

Appendix E: QRC performance using second order moments

  • Strong measurement erases two-qubit x- and y-direction correlations, while z components continue to follow the target.The reported explanation is that reservoir dynamics are mostly aligned with the z direction.Source: Appendix E · p. 14
  • RWP and RSP reach the unperturbed two-qubit STM capacity with 1.5·10^6 measurements, while weak online measurements preserve the short-delay plateau but not the longer-delay memory tail.The unperturbed two-qubit memory tail decays more slowly than for single-qubit observables.Source: Appendix E · p. 14
  • At g = 10, OLP memory decays exponentially with delay, whereas g = 0.5 preserves the unperturbed plateau only at small delays.With infinite measurements, performance improves, but strong g = 10 back-action still affects memory at longer delays; adding Gaussian noise recovers finite-sample OLP behavior.Source: Appendix E · pp. 14–15
  • For two-qubit observables, weak OLP measurements at g = 0.5 require nearly an order of magnitude more measurements to approach saturation than the single-qubit case.The weak OLP capacity rises more steeply than at g = 10, potentially reaching ideal sum capacity with fewer measurements.Source: Appendix E · p. 15
  • RSP is very inefficient in experimental time relative to RWP, and OLP requires more time than RWP to reach the same sum capacity.For weak g = 0.5, back-action is strong enough to be detrimental to performance.Source: Appendix E · p. 15

Appendix F: Memory capacity C(τ = 1) and non-classical signatures of QRC

At delay τ = 1, the appendix compares online predictions from weak and strong measurements across observable directions. It also notes that a classical-limit comparison was not investigated, although linearly independent observables, including noncommuting ones, contribute to processing capability.

Appendix F: Memory capacity C(τ = 1) and non-classical signatures of QRC

  • At τ = 1, weak OLP measurements retain significant single-qubit contributions from x, y, and z, while strong measurements drastically reduce x and y contributions, especially y.The account attributes this to strong measurement destroying small coherences that can contribute memory.Source: Appendix F · p. 15
  • The paper does not investigate the model's classical limit, identified as dynamics unaffected by back-action with processing along a fixed direction and no quantum coherence.It reports that linearly independent observables, including noncommuting operators, contribute to total processing capability in recognition and forecasting.Source: Appendix F · pp. 15–16

Appendix G: Experimental time resources and derivation of equation (15)

The appendix expresses experimental time for RSP, RWP, and OLP in terms of processing, measurement, reset, and washout costs. Its comparison relates measurement strength to washout steps under stated long-sequence and fast-measurement assumptions.

Appendix G: Experimental time resources and derivation of equation (15)

  • Experimental-time expressions for RSP, RWP, and OLP account for measurement time τM and reset time τR alongside processing time.The two-qubit expression includes a factor of 3 for separate measurements in x, y, and z; the main text assumes τM = τR = 0.Source: Appendix G · p. 16
  • The derivation assumes Nwo ≪ Nt, τM ≪ ∆t, and τR ≪ τwo to simplify the protocol comparison.The first washout-time term is neglected when the number of time steps is sufficiently large.Source: Appendix G · p. 16
  • For equal statistical error, the required measurement counts depend on protocol and measurement strength, with separate relations given for single- and two-qubit observables.The OLP strength must be weak enough to preserve exploitable memory yet strong enough to extract information and reduce measurement requirements.Source: Appendix G · pp. 16–17

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