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Optimal quantum learning of a unitary transformation

A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, P. Perinotti

arXiv:0903.0543v2quant-ph

TL;DR

The paper asks how to learn an unknown unitary from finite black-box uses so it can later be reproduced. For group-drawn unitaries, it shows that parallel storage followed by measuring the memory and applying the estimate is optimal, and extends the result to unitary inversion.

  • Problem

    The paper asks how to optimally learn an unknown unitary from finite black-box uses so it can later be reproduced on an input unknown at storage time.

    Source: §I · p. 1

  • Method

    For group-drawn unitaries, the strategy stores the available uses in parallel on a suitable entangled state, then estimates U from memory and applies the estimate M times.

    Source: §III · p. 3

  • Results

    For group-drawn unitaries, optimal retrieval uses measurement of the memory followed by applying the estimated unitary, and for M = 1 learning fidelity equals estimation fidelity.

    Source: A. The M = 1 case · p. 5

  • Takeaways & Limitations

    The result also gives optimal approximate realignment of reference frames for quantum communication when inverting an unknown unitary.

    Source: §I · p. 2, §V · p. 6

Abstract

from arXiv · show

We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine), and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a "measure-and-rotate" retrieving. Differing from the case of quantum cloning, where the incoherent "measure-and-prepare" strategies are typically suboptimal, in the case of learning the "measure-and-rotate" strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the "measure-and-rotate" strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.

I. INTRODUCTION

The paper frames unitary learning as using finite black-box calls to store an unknown transformation and later reproduce it. For group-drawn unitaries, it finds that estimation followed by applying the estimate can achieve optimal learning performance.

I. INTRODUCTION

  • A quantum memory could store an unknown transformation so it can be transmitted by sending a state rather than transferring the device.The stored transformation could also be processed using state-manipulation techniques.Source: §I · p. 1
  • Quantum learning uses N black-box calls to an unknown unitary now so it can be reproduced later on an input unknown at the time of storage.The user may be asked to produce M ≥ 1 copies, making performance decay with the required number of copies relevant.Source: §I · p. 1
  • The general learning problem is to optimize both the storing board and the retrieving channel, with parallel and sequential storage among the possible strategies.The most general storing strategy is a quantum circuit board with open slots for the input copies.Source: §I · p. 1
  • Estimation-based retrieval avoids quantum-memory costs and permits unlimited reproductions at constant quality, although analogous strategies are typically suboptimal for quantum cloning.The authors contrast that expectation with their finding that estimation-based strategies attain optimal unitary-learning performance for group-drawn transformations.Source: §I · pp. 1–2
  • For group-drawn unitaries, incoherent estimation-based strategies achieve optimal learning performance, including when only one output copy is required.For a completely unknown qubit unitary, the optimal fidelity behaves as F = 1 − O(N^-2) for large N.Source: §I · p. 2

II. NOTATION AND THEORETICAL FRAMEWORK

The framework represents channels and quantum networks through Choi operators and link products. It uses these representations to express circuit-board constraints as linear equations on positive operators.

II. NOTATION AND THEORETICAL FRAMEWORK

  • The Choi operator represents a channel as a positive operator, and channel composition is represented by the link product.A channel is completely positive and trace-preserving; trace preservation is expressed by a normalization condition.Source: §II · p. 2
  • Multipartite channels can be connected by matching output and input Hilbert-space labels, with the resulting Choi operators contracted using the link product.The example connects the K output of one channel to the K input of another.Source: §II · p. 2
  • Link products are commutative up to reordering Hilbert spaces, but channel composition is not; labels encode which output is connected to which input.Reordering spaces with distinct labels is safe, while composition order is reflected by the chosen input-output labels.Source: §II · pp. 2–3
  • A quantum comb is the Choi operator of a quantum circuit board, obtained by taking the link product of its component channels.The circuit board's sequence of trace-preserving channels is represented by linear constraints, so optimizing the board becomes optimization over a positive operator subject to those constraints.Source: §II · p. 3

III. OPTIMIZATION OF LEARNING

For a unitary drawn from a group, optimal learning uses parallel storage followed by an optimal estimation measurement and conditional application of the estimated unitary. For one output copy, both the optimal storage state and fidelity coincide with those of unitary estimation.

III. OPTIMIZATION OF LEARNING

  • The figure of merit averages output-state fidelity over pure inputs and unitaries drawn from the group.The optimization is equivalent, up to constants, to maximizing the channel fidelity averaged over the unknown unitary.Source: A. The M = 1 case · p. 3
  • Every storing board can be simulated by applying the N unitary examples in parallel to a suitable input state.The optimal input state can be chosen with components indexed by irreducible representations in the decomposition of U⊗N.Source: A. The M = 1 case · pp. 3–4
  • Optimal retrieval measures the memory with the optimal POVM and applies the estimated unitary to the new input.The stored examples are applied in parallel, and the optimal storage state has the same form as an optimal estimation state.Source: §III · p. 3, A. The M = 1 case · p. 5
  • For one output copy, the optimal storage state and learning fidelity are identical to those for estimating the unknown unitary.This lets the learning problem use known optimal estimation states and fidelities.Source: A. The M = 1 case · p. 5
  • For unknown SU(2) qubit unitaries, the large-N fidelity is approximately 1−π²/4N², while for an unknown phase shift it is approximately 1−2π²/(N+1)².The phase-shift example's optimal state can be prepared using only N qubits.Source: A. The M = 1 case · p. 5

B. Generalization to the M > 1 case

For reproducing more than one output copy, the paper distinguishes single-copy and global fidelity as separate criteria. It examines both notions of optimality.

B. Generalization to the M > 1 case

1. Optimal learning according to the single-copy fidelity

For single-copy fidelity, the local output channel is obtained by fixing the other inputs and tracing them out. The optimal measure-and-prepare strategy also optimizes these local channels, with fidelity that does not decrease as the number of outputs increases.

1. Optimal learning according to the single-copy fidelity

  • C(1)_U,Σ describes the first output when the remaining M − 1 inputs are prepared in state Σ.It is formed by applying C_U to the first input together with Σ, then tracing out all systems except the first.Source: §1 · p. 5
  • The fidelity of any such local channel with U cannot exceed the optimal estimation fidelity F_est.The same bound applies to each local channel C(i)_U,Σ, in which all inputs except the i-th are discarded.Source: §1 · p. 5
  • The measure-and-prepare strategy is optimal for maximizing every local channel’s single-copy fidelity, which does not decrease as M increases.Each local channel retains one output and discards the others.Source: §1 · p. 5

2. Optimal learning according to the global fidelity

For global fidelity across M outputs, optimal learning is equivalent to estimating the unknown group unitary and conditionally applying its M-fold output. The result also holds for nonidentical group representations.

2. Optimal learning according to the global fidelity

  • The optimal global-fidelity strategy stores the examples in parallel and measures the optimal POVM before conditionally applying the estimate as U_hat^⊗M.The corresponding estimation goal is f_M(U,U_hat) = (|Tr[U†U_hat]|/d)^2M.Source: §2 · p. 6
  • The coefficients p_j in the optimal input state generally depend on M.Source: §2 · p. 6
  • The argument extends to nonidentical input or output uses, provided each unitary belongs to a representation of the group G.For example, SO(3) examples may be rotations of particles with different angular momenta.Source: §2 · p. 6

IV. OPTIMAL INVERSION OF AN UNKNOWN UNITARY EVOLUTION

The inversion task replaces reproducing U with reproducing U†, and its optimal retrieval measures the stored examples and conditionally applies the estimated inverse. The fidelity-optimal input differs from the likelihood-optimal state, with distinct fidelity and success-probability behavior.

IV. OPTIMAL INVERSION OF AN UNKNOWN UNITARY EVOLUTION

  • Optimal inversion measures the optimal POVM on the stored state and conditionally applies U_hat†⊗M, yielding the optimal approximate reference-frame realignment strategy.In the communication setting, Alice’s reference-frame token accompanies Bob’s quantum message, which he tries to retrieve despite the frame mismatch.Source: §IV · p. 6
  • For M = 1 and G = SU(2), U(1), the fidelity-optimal state gives average fidelity approaching 1 as 1/N^2, while the likelihood-maximizing state scales as 1/N.Source: §IV · p. 6
  • For M = 1, the likelihood-maximizing state permits perfect correction with probability p = 1 − 3/(N + 1), whereas the fidelity-optimal state does not.The paper leaves the best state for maximizing success probability and the fidelity–probability trade-off as open problems.Source: §IV · p. 6

V. CONCLUSIONS

The paper establishes optimal storing and retrieval for unknown group transformations, with the incoherent measure-and-rotate strategy optimal even in contrast to quantum cloning. It extends this result to inversion and approximate reference-frame alignment, while leaving non-group unitaries for future study.

V. CONCLUSIONS

  • For unknown group transformations with N input and M output copies, the incoherent measure-and-rotate strategy is optimal, unlike the typical quantum-cloning case.Source: §V · p. 6
  • The inversion result applies to optimal approximate alignment of reference frames for quantum communication.Source: §V · p. 6
  • Whether coherent strategies improve learning when the unknown unitaries do not form a group remains open.The paper gives the unitary in Grover’s quantum search algorithm as an example of this setting.Source: §V · p. 6

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