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arXiv:2102.00740 · Quantum

Entanglement-Free Parameter Estimation of Generalized Pauli Channels

Junaid ur Rehman, Hyundong Shin

Published Jul 1, 2021• 19 citations
arXivHTMLPDF

Abstract

We propose a parameter estimation protocol for generalized Pauli channels acting on d -dimensional Hilbert space. The salient features of the proposed method include product probe states and measurements, the number of measurement configurations linear in d , minimal post-processing, and the scaling of the mean square error comparable to that of the entanglement-based parameter estimation scheme for generalized Pauli channels. We also show that while measuring generalized Pauli operators the errors caused by the Pauli noise can be modeled as measurement errors. This makes it possible to utilize the measurement error mitigation framework to mitigate the errors caused by the generalized Pauli channels. We use this result to mitigate noise on the probe states and recover the scaling of the noiseless probes, except with a noise strength-dependent constant factor. This method of modeling Pauli channel as measurement noise can also be of independent interest in other NISQ tasks, e.g., state tomography problems, variational quantum algorithms, and other channel estimation problems where Pauli measurements have the central role.

DOI
10.22331/q-2021-07-01-490

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We propose a parameter estimation protocol for generalized Pauli channels acting on d -dimensional Hilbert space. The salient features of the proposed method include product probe states and measurements, the number of measurement configurations linear in d , minimal post-processing, and the scaling of the mean square error comparable to that of the entanglement-based parameter estimation scheme for generalized Pauli channels. We also show that while measuring generalized Pauli operators the errors caused by the Pauli noise can be modeled as measurement errors. This makes it possible to utilize the measurement error mitigation framework to mitigate the errors caused by the generalized Pauli channels. We use this result to mitigate noise on the probe states and recover the scaling of the noiseless probes, except with a noise strength-dependent constant factor. This method of modeling Pauli channel as measurement noise can also be of independent interest in other NISQ tasks, e.g., state tomography problems, variational quantum algorithms, and other channel estimation problems where Pauli measurements have the central role.
Optimal probe states and measurements are always required to achieve the best estimation precision in quantum metrology. In the actual physical environment, ubiquitous noise hinders estimation precision and amplifies the challenges of optimizing the probe states and measurements. In this study we extend a theoretical proposal, presented in Phys. Rev. A 103, 032615 (2021), to encompass noisy general Pauli channels for SU(2) phase estimation. By utilizing a superposition of different causal orders of two channels, we establish theoretically a probe-state-independent criterion for the estimation precision. We show that the probe-state-independent property results from the noncommutativity of the Kraus operators of the parameter-encoding channels. Based on this criterion, one can find some parameter-encoding channels without the requirement of precisely preparing the probe state. Moreover, our scheme requires only deterministic projection measurement on the control qubit. In this way, one can simultaneously avoid optimizing both the probe states and measurements. In addition, we also show that the estimation precision and probe-state-independent property of the indefinite causal order scheme are independent of representations of the Kraus operators. This alleviates the experimental challenges in realizing quantum channels. We demonstrate experimentally the advantages of the indefinite causal order for phase estimation of an SU(2) unitary transformation in the three kinds of quantum noise channels. Our result shows that the dynamic evolution of indefinite causal order can outperform the conventional cascaded estimation scenario at high noise levels, exhibiting high robustness and feasibility in practical estimation tasks.
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here, we prove several results on efficiently learning Pauli channels and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits with high probability to a relative precision ϵ using O (ϵ -2 n2 n ) measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors, which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next, we show that the error rates for an arbitrary set of s Pauli errors can be estimated to a relative precision ϵ using O (ϵ -4 log s log s/ϵ) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k -local correlations, we can learn an entire n -qubit Pauli channel to relative precision ϵ with only O k (ϵ -2 n 2 log n ) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.
We establish the necessary and sufficient conditions for local unbiased estimation in multi-parameter estimation tasks. More specifically, we first consider quantum state estimation, where multiple parameters are encoded in a quantum state, and derive simple and intuitive necessary and sufficient conditions for a local unbiased estimation based on the derivatives of the encoded state. To demonstrate the utility of our framework, we consider phase estimation under unknown Pauli noise. We show that while local unbiased phase estimation is infeasible with a naive scheme, employing an entangled probe with a noiseless ancilla enables local unbiased estimation. We then extend our analysis to quantum channel estimation and derive a necessary and sufficient condition for local unbiased estimability of channel parameters, allowing noiseless ancillae, general CPTP controls, and multiple uses of the channel. As a concrete application, we analyze unbiased estimation of noise parameters affecting non-Clifford gates via cycle benchmarking under SPAM errors.
We consider the problem of estimating time-varying quantum noise. Specifically, we focus on Pauli qubit noise with time-variations and attempt to construct the most accurate instantaneous channel description. To this end, we propose an adaptive framework of simultaneous communication and parameter estimation (SCAPE) that efficiently and accurately estimates the time-varying Pauli channel while communicating reliably over the channel being estimated. This adaptive framework gives the informed control of communication rate–parameter estimation tradeoff to communicating parties. Interestingly, this adaptive SCAPE requires post-processing entirely on the receiver’s end and minimal feedback to the sender to increase, decrease, or continue with the same code rate of employed error correcting code. This procedure can be particularly useful in time-varying quantum channels with natural periodic deviations in channel conditions, e.g., in satellite communication channels.
Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n -qubit channel to precision ϵ in ℓ∞ using just O(1/ϵ2)log⁡(n/ϵ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O(1/ϵ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤1/4 .We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1−η . In the regime of small η we extend our algorithm to achieve multiplicative precision 1±ϵ (i.e., additive precision ϵη ) using just O(1ϵ2η)log⁡(n/ϵ) applications of the channel.
We propose a scheme for simultaneous classical communication and parameter estimation of generalized Pauli channels. This framework relies on the facts that i) under certain conditions, Pauli channels act as classical symmetric channel, and ii) a recently proposed parameter estimation method of Pauli channels can be executed while satisfying the conditions of i). The performance of the parameter estimation part in this simultaneous scheme relies on the symbol error rate of the communication module. This reliance gives rise to a natural tradeoff between the communication rate and the performance of parameter estimation in this simultaneous scheme. We exemplify our protocol by utilizing repetition codes for communication and the aforementioned parameter estimation scheme for the characterization of the channel at hand. We explore the tradeoff behavior between the communication rate and the performance of parameter estimation in the numerical examples. Our proposal paves the way for practical methods in simultaneous communication and channel estimation over unknown quantum channels.