Self-Guided Quantum Tomography

Abstract

We introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography (SGQT) uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that SGQT is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of post-processed state estimation.The act of inferring a quantum mechanical description of a physical system-assigning it a quantum state-is referred to as tomography. Tomography is a required, and now routine, task for designing, testing and tuning qubits-the building blocks of a quantum information processing device [1]. However, in a grand irony, the exact same exponential scaling that gives a quantum information processing device its power also limits our ability to characterize it.That tomography is a problem exponentially hard in the number of qubits has lead to many proposals for efficient learning within restricted subsets of quantum states [2,3]. On the other hand, if we expect to have prepared a specific target state, efficient protocols exist to estimate the fidelity to this state [4,5]. These protocols are direct in the sense that few measurements are required to provide an estimate of the fidelity to the target rather than first reconstructing the state then calculating the fidelity.The proposal presented here is direct in the same sense as [4,5], but converges to the state itself. The algorithm is iterative-from directly estimating a distance measure to the underlying state, the experiment guides itself to a description of its own state: self-guided quantum tomography (SGQT). The distance measure m is arbitrary, in the sense that any measure will work. However, the more rapidly the experiment can provide an estimate for m, the more rapidly SGQT will converge, such that if m can be estimated efficiently, then SGQT will be efficient.Before describing exactly what SGQT is, we first state what it is not by reviewing the problem of tomography. There is some true state ρ which generates data, a list of measurement outcomes corresponding to effects D = {E 0 , E 1 , . . .}. The probability to observe this data is given by the Born ruleThe prevailing method is to solve the inverse problem of identifying an accurate estimate, σ, of ρ given a sample data set drawn from this distribution. Here the approach is quite different. We begin with a distance measure on states m(ρ, σ). The only requirement is that this measure can be estimated from experiment, such that we have access toThis quantity fluctuates from noise which can come from a variety of sources but is always present due to the fundamental statistical nature of quantum mechanics-also known as shot-noise.Here we will provide an algorithm to iteratively propose new states σ such that we converge to ρ only via estimates of f (σ). The core of the algorithm is a stochastic optimizatio...

Scalable on-chip quantum state tomography

“…Some approaches seek to avoid unnecessary measurements by assuming that the system is in particular low rank states, such as sparse states [14,15] or low dimensional matrix product states [16]. Alternatively tomography can be 'self-guided', where real-time feedback of measurement results guides the next choice of the measurement basis [17,18], helping to avoid taking measurements that have limited utility for analyzing the state. It has been shown that tomography procedures involving some global quantum measurements have increased error robustness relative to using only local qubit measurements, and thus can be completed in less time [19].…”

“…scales exponentially, and processing of measurement data can be extremely resource demanding. We employ an optimization technique known as the simultaneous perturbation stochastic approximation [31], similar to the algorithm formulated for self-guided quantum state tomography [17]. It minimizes a distance measure between the true state and the algorithm's current guess.…”

“…The algorithm for reconstructing the density matrix from a measured set of correlations uses the simultaneous perturbation stochastic approximation (SPSA) [31]. We follow an optimization process very similar to [17], except we adapt it to work with mixed quantum states. This is achieved by defining the density matrix as, ρ(t) =T † (t)T (t)/Tr{T † (t)T (t)}, as in Eq.…”

Adaptive identification of coherent states

“…Generally, the specific set of rules, also called a policy, according to which the measurement setting s is to be updated, has to be developed separately for each problem at hand. This is true also for the recently developed technique called self-guided quantum tomography (SGQT) [56]. SGQT searches the estimate of a quantum state by making measurements in the directions close to the estimate, approaching it as a power law as a function of adaptive iteration steps.…”

Optimized tomography of continuous variable systems using excitation counting

“…However, QST is often resource-consuming, involving preparation of a large number of identical unknown states and measurement of a large set of independent observables. For qubit systems, many techniques have been developed to reduce the cost of full state tomography, such as compressed sensing [1][2][3], permutationally invariant tomography [4], self-guided/adaptive tomography [5,6], matrix product states tomography [7]. In contrast, for continuous variable (CV) systems that also play an important role in quantum information, the standard techniques in use today are decades old, namely homodyne measurement [8,9] for optical photons and direct Wigner function measurement [10][11][12] for cavity QED.…”