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arXiv:1102.3700 · Physical Review A

Characterization of a qubit Hamiltonian using adaptive measurements in a fixed basis

Alexandr Sergeevich, Anushya Chandran, Joshua Combes, Stephen D. Bartlett, Howard M. Wiseman

Published Nov 15, 2011• 78 citations
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Abstract

We investigate schemes for Hamiltonian parameter estimation of a two-level system using repeated measurements in a fixed basis. The simplest (Fourier based) schemes yield an estimate with a mean-square error (MSE) that decreases at best as a power law $\ensuremath{\sim}{N}^{\ensuremath{-}2}$ in the number of measurements $N$. By contrast, we present numerical simulations indicating that an adaptive Bayesian algorithm, where the time between measurements can be adjusted based on prior measurement results, yields a MSE which appears to scale close to $\mathrm{exp}(\ensuremath{-}0.3N)$. That is, measurements in a single fixed basis are sufficient to achieve exponential scaling in $N$.

DOI
10.1103/physreva.84.052315

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We investigate schemes for Hamiltonian parameter estimation of a two-level system using repeated measurements in a fixed basis. The simplest (Fourier based) schemes yield an estimate with a mean-square error (MSE) that decreases at best as a power law \ensuremath∼N\ensuremath−2\ensuremath{\sim}{N}^{\ensuremath{-}2}\ensuremath∼N\ensuremath−2 in the number of measurements NNN. By contrast, we present numerical simulations indicating that an adaptive Bayesian algorithm, where the time between measurements can be adjusted based on prior measurement results, yields a MSE which appears to scale close to exp(\ensuremath−0.3N)\mathrm{exp}(\ensuremath{-}0.3N)exp(\ensuremath−0.3N). That is, measurements in a single fixed basis are sufficient to achieve exponential scaling in NNN.
We consider a two-mode bosonic state with fixed photon number n∈Nn \in \mathbb {N}n∈N , whose upper and lower modes pick up a phase ϕ\phi ϕ and −ϕ-\phi −ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating ϕ\phi ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider ϕ\phi ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider n=1n=1n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
Using Bayesian experimental design techniques, we have shown that for a single twolevel quantum mechanical system under strong (projective) measurement, the dynamical parameters of a model Hamiltonian can be estimated with exponentially improved accuracy over offline estimation strategies. To achieve this, we derive an adaptive protocol which finds the optimal experiments based on previous observations. We show that the risk associated with this algorithm is close to the global optimum, given a uniform prior. Additionally, we show that sampling at the Nyquist rate is not optimal.
Adaptive data collection and analysis, where data are being fed back to update the measurement settings, can greatly increase speed, precision, and reliability of the characterization of quantum systems. However, decoherence tends to make adaptive characterization difficult. As an example, we consider two coupled discrete quantum systems. When one of the systems can be controlled and measured, the standard method to characterize another, with an unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$, is swap spectroscopy. Here, adapting measurements can provide estimates whose error decreases exponentially in the number of measurement shots rather than as a power law in conventional swap spectroscopy. However, when the decoherence time is so short that an excitation oscillating between the two systems can only undergo less than a few tens of vacuum Rabi oscillations, this approach can be marred by a severe limit on accuracy unless carefully designed. We adopt machine learning techniques to search for efficient policies for the characterization of decohering quantum systems. We find, for instance, that when the system undergoes more than 2 Rabi oscillations during its relaxation time T1,O(103){T}_{1}, O({10}^{3})T1​,O(103) measurement shots are sufficient to reduce the squared error of the Bayesian initial prior of the unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$ by a factor O(104)O({10}^{4})O(104) or larger. We also develop policies optimized for extreme initial parameter uncertainty and for the presence of imperfections in the readout.
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a kkk-body Hamiltonian HHH whose correlation matrix has a spectral gap ΔΔΔ can be estimated to precision ε\varepsilonε with only O~(n3k/(εΔ)3/2)\tilde{O}\bigl(n^{3k}/(\varepsilon Δ)^{3/2}\bigr)O~(n3k/(εΔ)3/2) measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of HHH starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of mmm Pauli operators with weight ≤k\le k≤k to precision εεε using only O(ε−23klog⁡m)O(ε^{-2} 3^k \log m)O(ε−23klogm) measurements, which quadratically improves a recent result by Cotler and Wilczek.
We compare the accuracy, precision, and reliability of different methods for estimating key system parameters for two-level systems subject to Hamiltonian evolution and decoherence. It is demonstrated that the use of Bayesian modeling and maximum likelihood estimation is superior to common techniques based on Fourier analysis. Even for simple two-parameter estimation problems, the Bayesian approach yields higher accuracy and precision for the parameter estimates obtained. It requires less data, is more flexible in dealing with different model systems, can deal better with uncertainty in initial conditions and measurements, and enables adaptive refinement of the estimates. The comparison results show that this holds for measurements of large ensembles of spins and atoms limited by Gaussian noise as well as projection noise limited data from repeated single-shot measurements of a single quantum device.