Foundations for learning from noisy quantum experiments

Abstract

Understanding what can be learned from experiments is central to scientific progress. In this work, we use a learning-theoretic perspective to study the task of learning physical operations in a quantum machine when all operations (state preparation, dynamics, and measurement) are a priori unknown. We prove that, without any prior knowledge, if one can explore the full quantum state space by composing the operations, then every operation can be learned. When one cannot explore the full state space but all operations are approximately known and noise in Clifford gates is gate-independent, we find an efficient algorithm for learning all operations up to a single unlearnable parameter characterizing the fidelity of the initial state. For learning a noise channel on Clifford gates to a fixed accuracy, our algorithm uses quadratically fewer experiments than previously known protocols. Under more general conditions, the true description of the noise can be unlearnable; for example, we prove that no benchmarking protocol can learn gate-dependent Pauli noise on Clifford+T gates even under perfect state preparation and measurement. Despite not being able to learn the noise, we show that a noisy quantum computer that performs entangled measurements on multiple copies of an unknown state can yield a large advantage in learning properties of the state compared to a noiseless device that measures individual copies and then processes the measurement data using a classical computer. Concretely, we prove that noisy quantum computers with two-qubit gate error rate 𝜖 can achieve a learning task using 𝑁 copies of the state, while 𝑁 Ω(1/𝜖) copies are required classically.

The complexity of NISQ

“…While a single run of any noiseless bounded-depth circuit cannot achieve Grover speedup 65 , hybrid quantum-classical algorithms can perform many runs of noiseless bounded-depth circuits that depend adaptively on previous measurement outcomes. We prove that Grover speedup is impossible with hybrid quantum-classical algorithms using tools developed recently in the context of lower bounds for learning quantum states and processes using adaptive single-copy measurements [47][48][49][50][66][67][68] .…”

“…On the other hand, for small noise rate λ, the exponential scaling in the lower bound for NISQ algorithms has a base which is close to one. In 49 it was shown that NISQ algorithms can achieve (1−λ) −Θ(n) even for more general noise models. This suggests that NISQ algorithms can still perform well for learning quantum systems with a few hundred qubits 50 .…”

“…The excitement around NISQ algorithms has led to a plethora of new near-term algorithms targeting different applications, including quantum chemistry 3,28,[31][32][33] , machine learning 17,[34][35][36][37][38][39] , combinatorial optimization [40][41][42][43] , linear system solvers [44][45][46] , and experimental data analysis [47][48][49][50][51][52] . However, to the best of our knowledge, no existing works have rigorously examined the class of all possible NISQ algorithms and studied their inherent computational power.…”

The learnability of Pauli noise

“…Recently, Ref. [31] considered similar issues of noise learnability. They studied a different Pauli noise model with perfect initial state |0 , perfect computational basis measurement, and noisy single qubit gates, and showed the existence of unlearnable information.…”

“…This proof naturally implies that using other noisy gates from the gate set (that are subject to independent unknown noise channels) does not change the learnability of Pauli fidelities. On the other hand, it is known that under the stronger assumption of gate-independent noise (where different multi-qubit gates are assumed to have the same noise channel), the noise channel is fully learnable [29][30][31].…”