PaperlayerView exact revision on arXiv

Source-linked AI summary

Temporal Information Processing on Noisy Quantum Computers

Jiayin Chen, Hendra I. Nurdin, Naoki Yamamoto

arXiv:2001.09498v2quant-phcs.SYeess.SYstat.ML

TL;DR

Earlier quantum-reservoir proposals were difficult to implement on gate-model devices, while NISQ applications had largely focused on static tasks. This paper proposes dissipative quantum reservoirs, proves their class universal for nonlinear fading-memory maps, and demonstrates small noisy reservoirs performing nonlinear temporal tasks.

  • Problem

    Earlier quantum-reservoir proposals were challenging to realize on gate-model devices because their dynamics required many Trotterized gates, while NISQ demonstrations included static classification and quantum chemistry.

    Source: §I · p. 2

  • Method

    The authors use dissipative quantum dynamics as fixed reservoirs with trainable readouts, and describe an implementable subclass using current gate-model hardware.

    Source: §I · pp. 1–2, §IV · p. 3, §V · p. 5

  • Results

    The reservoir class uniformly approximates any nonlinear fading-memory map, and proof-of-principle experiments show small noisy reservoirs tackling complex nonlinear temporal tasks.

    Source: §I · p. 2

  • Takeaways & Limitations

    The approach offers potential for time-series modeling and signal processing on near-term gate-model quantum devices without quantum error correction.

    Source: §I · p. 2, §VII · p. 9

    The hardware-noise invariance statement assumes stationary Markovian noise, although experimental hardware noise could be time-varying.

    Source: B. Invariance under stationary… · p. 4

Abstract

from arXiv · show

The combination of machine learning and quantum computing has emerged as a promising approach for addressing previously untenable problems. Reservoir computing is an efficient learning paradigm that utilizes nonlinear dynamical systems for temporal information processing, i.e., processing of input sequences to produce output sequences. Here we propose quantum reservoir computing that harnesses complex dissipative quantum dynamics. Our class of quantum reservoirs is universal, in that any nonlinear fading memory map can be approximated arbitrarily closely and uniformly over all inputs by a quantum reservoir from this class. We describe a subclass of the universal class that is readily implementable using quantum gates native to current noisy gate-model quantum computers. Proof-of-principle experiments on remotely accessed cloud-based superconducting quantum computers demonstrate that small and noisy quantum reservoirs can tackle high-order nonlinear temporal tasks. Our theoretical and experimental results pave the path for attractive temporal processing applications of near-term gate-model quantum computers of increasing fidelity but without quantum error correction, signifying the potential of these devices for wider applications including neural modeling, speech recognition and natural language processing, going beyond static classification and regression tasks.

I. INTRODUCTION

The paper proposes quantum reservoir computers for nonlinear temporal processing on noisy gate-model quantum devices. It establishes universality for fading-memory maps and reports proof-of-principle experiments on superconducting processors.

I. INTRODUCTION

  • The proposed approach uses dissipative quantum dynamics as reservoirs to process input sequences into output sequences on gate-model NISQ computers.Reservoir computing maps sequential inputs into a high-dimensional state space and trains a simple linear readout.Source: §I · p. 1
  • The proposed quantum reservoirs form a universal class for approximating nonlinear fading-memory input-output maps arbitrarily closely and uniformly over input sequences.Universality belongs to the class, not to any individual reservoir.Source: §I · p. 2
  • Earlier quantum-reservoir proposals faced a gate-model implementation challenge because their dynamics required many gates through Trotterization.A prior variation had established universality for nonlinear fading-memory maps, but implementation on quantum gate-model hardware remained difficult.Source: §I · p. 2
  • The work also presents proof-of-principle experiments showing that small, noisy quantum reservoirs can tackle complex nonlinear temporal tasks without error mitigation.The experiments use remotely accessed IBM superconducting processors without quantum error correction.Source: §I · p. 2
  • The paper proposes a universal quantum-reservoir class and an implementable subclass using fixed circuits and native gates without precise gate-parameter tuning.The subclass is designed for current NISQ devices.Source: §I · p. 2

II. TEMPORAL INFORMATION PROCESSING

This section frames temporal processing as learning input-output relationships between sequences. It presents multi-step prediction and map emulation as two tasks and focuses on maps with fading memory.

II. TEMPORAL INFORMATION PROCESSING

  • An input-output map relates infinite input sequences to infinite output sequences, with the output at each time indexed by the map.Such maps can be realized by convergent dynamical systems that forget their initial condition.Source: §II · p. 2
  • Multi-step-ahead prediction trains on an initial sequence of input-output pairs and uses the learned map to predict later outputs.The stated training data exclude the first four timesteps to remove transient response.Source: §II · p. 2
  • Map emulation trains on multiple input-output sequences so the learned map can approximate outputs for a previously unseen input sequence.The request describes K training pairs and evaluates a further input sequence.Source: §II · p. 2
  • For a fading-memory map, outputs become increasingly less dependent on inputs from much earlier times.The paper targets nonlinear fading-memory maps using quantum reservoir computers.Source: §II · p. 2

III. RESERVOIR COMPUTING

Reservoir computing uses a fixed dynamical system to transform sequential inputs into states, then trains a readout to produce outputs. The reservoir must forget its initial condition, yielding a fading-memory input-output map.

III. RESERVOIR COMPUTING

  • A reservoir transforms each input into a state using arbitrary but fixed dynamics, while tunable parameters appear in the readout function.The reservoir dynamics are not trained; the required property is echo-state or convergence behavior that asymptotically forgets the initial condition.Source: §III · p. 3
  • After the initial-condition effect washes out, linear regression optimizes readout parameters to minimize empirical mean squared error.The resulting input-output map is considered to have fading memory.Source: §III · p. 3
  • Classical reservoir-computing studies report state-of-the-art chaotic-system modeling and hardware uses for real-time processing, with state dimension relevant to some tasks.The cited examples include spoken-digit recognition.Source: §III · p. 3

IV. UNIVERSAL QUANTUM RESERVOIR COMPUTERS

The paper defines quantum reservoirs whose dynamics and readouts yield fading-memory maps, and proves the class universal for nonlinear fading-memory maps. It also describes a subclass based on fixed, nontrivial quantum dynamics and a polynomial readout.

IV. UNIVERSAL QUANTUM RESERVOIR COMPUTERS

  • A quantum reservoir comprises non-interacting subsystems whose state evolves through input-dependent quantum dynamics and fixed density operators.The dynamics are convergent and asymptotically forget the initial reservoir state.Source: §IV · p. 3
  • The reservoir uses fixed, nontrivial CPTP maps without requiring precise tuning or engineering of those maps.Examples include naturally occurring noisy channels such as dephasing or amplitude damping.Source: §IV · p. 3
  • The readout measures Pauli Z expectations and applies a degree-R multivariate polynomial, with a linear readout used in the proof-of-principle experiments.The readout parameters are tunable.Source: §IV · p. 3
  • For any nonlinear fading-memory map and any δ > 0, the universal quantum-reservoir class contains a member that approximates it uniformly over all inputs.Universality is a property of the class, not of an individual reservoir.Source: §IV · p. 3

A. A subclass implementable on noisy gate-model quantum devices

The paper defines a hardware-compatible subclass of universal quantum reservoirs for noisy gate-model devices. Its dynamics use fixed, nontrivial gates and encode input and forgetting through classical mixtures.

A. A subclass implementable on noisy gate-model quantum devices

  • The proposed subclass makes universal quantum-reservoir dynamics implementable on current gate-model devices despite hardware restrictions.The general universal-reservoir dynamics cannot all be efficiently implemented with limited qubits and other current hardware constraints.Source: A. A subclass implementable on noisy… · p. 4
  • Its fixed unitary gates may use native single- and multi-qubit gates, with parameters chosen arbitrarily but avoiding trivial dynamics.Precise parameter tuning is not required.Source: A. A subclass implementable on noisy… · p. 4
  • The input is encoded as a classical mixture, while each subsystem’s forgetting rate ϵ_k sets the probability of resetting its state.With probability ϵ_k, the subsystem state is replaced; otherwise it is unchanged with probability 1 − ϵ_k.Source: A. A subclass implementable on noisy… · p. 4

B. Invariance under stationary Markovian hardware noise and time-invariant readout error

The universal reservoir family remains universal under stationary Markovian noise because the noisy dynamics retain the reservoir form. Linear readout also leaves predicted outputs unchanged under time-invariant readout error.

B. Invariance under stationary Markovian hardware noise and time-invariant readout error

  • Stationary Markovian noise preserves the quantum-reservoir form, so the universal family remains universal under that noise model.For the implementable subclass, such hardware noise replaces the fixed unitaries while retaining the form of the reservoir dynamics.Source: B. Invariance under stationary… · p. 4
  • The hardware-noise invariance claim is conditional: experiments and simulator agreement indicate approximate preservation during the experiments, while the hardware noise could be time-varying.If noise changes slowly, periodically retraining output weights is described as possible, but remains challenging on cloud-accessed-only devices.Source: B. Invariance under stationary… · p. 4
  • With a linear readout, time-invariant readout error leaves predicted outputs unchanged after the output weights are optimized by linear regression.The error introduces a time-invariant linear transformation of the measurement data.Source: B. Invariance under stationary… · p. 4

V. REALIZATION OF A SUBCLASS ON CURRENT QUANTUM HARDWARE

The paper presents two implementations of a quantum-reservoir subclass, including a sampling scheme for devices with pure-state preparation and a QND scheme when reset is available. Convergence enables versions whose applications of the dynamics do not grow with input length.

V. REALIZATION OF A SUBCLASS ON CURRENT QUANTUM HARDWARE

  • The proof-of-principle implementation uses Monte Carlo sampling to realize mixed-state reservoir dynamics with pure-state preparation, reinitializing circuits to estimate each timestep.This first scheme is used because the IBM superconducting devices in the experiments do not provide qubit reset.Source: §V · p. 5
  • When qubit reset is available, QND measurements retain the encoded state across timesteps, reducing circuit runs from N_mSL to N_mS.This scheme requires N_mSL applications of the reservoir dynamics; the first scheme requires N_mSL circuit runs.Source: §V · p. 5
  • Using convergence, both schemes can estimate outputs from only the last M inputs, with N_mSM dynamics applications independent of input length L.The first scheme uses N_mSL circuit runs; the second uses N_mS circuit runs, and both are described as efficient versions.Source: §V · p. 5

VI. PROOF-OF-PRINCIPLE EXPERIMENTS

The experiments use five nonlinear tasks chosen to test different computational aspects of the quantum-reservoir proposal, and implement four reservoirs across three IBM superconducting processors.

VI. PROOF-OF-PRINCIPLE EXPERIMENTS

  • The five tasks probe high-dimensional nonlinear maps, highly nonlinear dynamics, short-term memory, and long-term memory beyond the theoretical guarantee.Tasks I and II use linear dynamics with different matrix sparsities and forgetting rates; Tasks I–IV have fading memory.Source: §VI · p. 5
  • Four single-subsystem reservoirs with linear readouts are implemented on three IBM superconducting processors, using 4-, 5-, and 10-qubit configurations.The 4- and 10-qubit reservoirs use Boeblingen; two distinct 5-qubit reservoirs use Ourense and Vigo.Source: §VI · p. 6
  • The four-reservoir comparison is designed to investigate the effects of reservoir size, circuit complexity, and intrinsic hardware noise on approximation performance.The devices differ in qubit couplings and gate-error characteristics.Source: §VI · p. 6

A. Quantum circuits for QRs

The experimental quantum reservoirs use two distinct circuit unitaries designed for nontrivial dynamics and hardware-efficient implementation. Their initialization also bypasses a conventional washout sequence by starting at a steady state.

A. Quantum circuits for QRs

  • Initializing the QR circuits in |0⟩^n bypasses washout because that state is steady under constant input u_l = 1.The design chooses the initial state and U0 so |0⟩^n is the steady state for that input.Source: A. Quantum circuits for QRs · p. 6
  • U0 and U1 are distinct, hardware-efficient circuits chosen to produce non-trivial quantum dynamics.Their layers act on different target-control qubit pairs, and the circuits use native IBM gates.Source: A. Quantum circuits for QRs · p. 6, Figure 2 · p. 6
  • The Boeblingen reservoirs use N0 = N1 = 5 layers, while the Ourense and Vigo reservoirs use different circuit forms.Vigo uses rotational X and Y gates; the excerpts do not specify the full circuit forms for each device.Source: A. Quantum circuits for QRs · p. 6

B. Experimental implementation

The experiments estimate QR states from repeated circuit runs and train separate readouts for multiple temporal tasks using shared reservoir dynamics. Evaluation uses randomly sampled inputs and proceeds without process or readout error mitigation.

B. Experimental implementation

  • Experiments use 1,024 circuits and 1,024 shots on Boeblingen, versus 8,192 shots for the 5-qubit QRs.The shot counts reflect the devices’ circuit execution times.Source: B. Experimental implementation · p. 7
  • Inputs are uniformly sampled from [0, 1] after a length-50 constant washout, with early transient data discarded.Training and test spans differ between multi-step prediction and map emulation, and available sequence lengths are limited by circuit-length constraints and shared cloud processors.Source: B. Experimental implementation · p. 7
  • The four QRs’ fixed dynamics and gate parameters are reused across tasks, while each task receives independently optimized readout parameters.The QR state estimates are recorded once for all time steps.Source: B. Experimental implementation · p. 7
  • The reported results use no process or readout error mitigation.Performance is evaluated using normalized mean-squared error.Source: B. Experimental implementation · p. 7

C. QR task performance

The 10-qubit Boeblingen QR performs strongly on the multi-step prediction tasks, while spatially multiplexing two 5-qubit QRs yields competitive map-emulation performance. Results vary by task, with Task V an exception to the Boeblingen QR’s advantage over the 5-qubit devices.

C. QR task performance

  • The Ourense and Vigo QRs have similar noise and the same coupling map, yet Vigo outperforms Ourense on every task, suggesting longer gate sequences may improve performance.The multiplexed QR combines computational features of the two 5-qubit devices.Source: C. QR task performance · pp. 7–8
  • NMSE < 0.08 for the 10-qubit Boeblingen QR, which outperforms smaller QRs on the first four multi-step tasks and nearly doubles performance on Tasks II and III.Two qubits with experimental deviations from simulation had their readout weights set to zero.Source: Table I · p. 7, C. QR task performance · p. 7
  • The spatially multiplexed pair achieves NMSEs 0.20, 0.13, 0.16, 0.25, and 0.20 across five map-emulation tasks, with near-twofold gains on Tasks II and III.The experiments used no readout or process error mitigation.Source: Table II · p. 7, C. QR task performance · p. 8
  • The 10-qubit Boeblingen QR outperforms the 5-qubit QRs on all tasks except Task V.The authors suggest higher Boeblingen noise or a target map not known to have fading memory could explain the exception.Source: C. QR task performance · p. 7

VII. CONCLUSION

The section summarizes universal quantum reservoirs for fading-memory temporal maps, their nonlinear temporal processing on noisy gate-model devices, and related results on scalability, readout-error invariance, and QND-based implementation. It also outlines runtime-reduction and output-estimation methods, plus the tasks, circuits, and sequential data used to specify experimental implementation and evaluation.

VII. CONCLUSION

  • Small-scale noisy quantum reservoirs performed non-trivial nonlinear temporal tasks under current hardware limitations, without readout or process-error mitigation.The proposed class can use arbitrary fixed circuits native to noisy processors without precise circuit-parameter tuning.Source: §VII · p. 8
  • The approach offloads exponentially costly computations to quantum systems while classical sequential-data processing costs scale linearly with qubit number.The reservoir’s microsecond evolution suggests potential for real-time signal processing; spatial multiplexing of small noisy reservoirs experimentally produced a computational boost.Source: §VII · pp. 8–9

1. The convergence property

  • The QR dynamics are convergent, so the reservoir forgets its initial condition as it processes inputs.The convergence lemma applies to the QR dynamics defined by the paper’s equations.Source: §1 · pp. 9–10

2. The fading memory property

  • For any null sequence w, the input-output map induced by the QR is w-fading memory.This result concerns the QR dynamics with the associated readout function.Source: §2 · p. 10

3. The universality property

  • Quantum reservoir computers can approximate any nonlinear fading-memory map arbitrarily closely and uniformly over inputs.The QR class is dense in the space of w-fading memory maps for any null sequence w.Source: §VII · p. 8, §3 · p. 10
  • The universality proof establishes that the QR map class is dense by showing it forms a polynomial algebra that separates input sequences.The proof invokes the Stone-Weierstrass theorem after establishing separation of points.Source: §3 · pp. 10–11

Appendix B: Invariance under time-invariant readout error

  • With a linear readout, QR predicted outputs are invariant under time-invariant readout error.The derivation assumes the calibration matrix’s pseudoinverse has linearly independent rows.Source: Appendix B · pp. 11–12

Appendix C: Efficient implementations of a subclass on gate-model quantum computers

  • With qubit reset and QND measurements, the second implementation scheme needs N_mSL applications of T(u_l) but only N_mS circuit runs, independent of sequence length L.The QND measurements retain information encoded in the reservoir state between timesteps.Source: Appendix C · p. 12
  • The convergence property reduces the input-dependent map applications needed by both quantum-reservoir schemes, with the second scheme’s map applications and circuit runs independent of sequence length.For the first scheme, map applications fall from quadratic dependence on sequence length L to at most NmSLM; the second requires at most NmSM map applications and NmS circuit runs.Source: Appendix C · p. 13

Appendix D: Monte Carlo estimation

  • Monte Carlo measurement averages are unbiased at finite sample sizes, while convergence-based truncation introduces a bias that can be suppressed exponentially by choosing M based on ǫ.The approximation compares estimates of ⟨g Z(i)⟩l and ⟨Z(i)⟩l.Source: Appendix D · pp. 13–14

A. Nonlinear temporal processing tasks

  • The five evaluation tasks span linear and nonlinear reservoir models, a Volterra series, and missile dynamics.Tasks I and II use linear reservoirs with degree-2 polynomial readouts and different memory settings; Task III is a classical reservoir model for chaotic-system modeling, Task IV is a Volterra series with kernel order 5 and memory 2, and Task V models a missile with constant horizontal-plane velocity as a continuous-time long-term-memory nonlinear map.Source: A. Nonlinear temporal processing tasks · p. 14

B. Quantum circuits for QRs

  • The proof-of-principle experiments use quantum-reservoir circuits for 4- and 10-qubit Boeblingen devices and 5-qubit Ourense and Vigo devices.The circuits are shown in Figs. 7 and 8, respectively.Source: B. Quantum circuits for QRs · p. 15

C. Full input-output sequential data

  • Experimental sequences bypass reservoir washout by initialization, discard four initial input-output points, and include full washout, training, and test sequences for both prediction and map emulation.The washout length is Lw = 50 for all tasks, with ul = 1 used for Task IV at l = −2, −1.Source: C. Full input-output sequential data · p. 15

D. Measurement and simulation data

Hardware experiments compared quantum reservoirs with ideal and calibration-based noisy simulations across multi-step prediction and map emulation. The results include a 10-qubit readout-error issue, while the hardware section describes the processors and circuits used.

E. Hardware specifications

  • The experiments used IBM superconducting processors: 20-qubit Boeblingen and 5-qubit Ourense and Vigo.For the Boeblingen circuits, the maximum execution duration was about 74.7 μs, within the coherence times of most selected qubits.Source: E. Hardware specifications · p. 15
  • Distinct Ourense and Vigo circuit designs were chosen to combine different computational features for spatial multiplexing.The 5-qubit circuits were simpler than the Boeblingen circuits because the devices had more restricted qubit couplings.Source: E. Hardware specifications · pp. 15–16, Figure 8 · p. 16

D. Measurement and simulation data

Loading 2001.09498v2…