Joint quantum-state and measurement tomography with incomplete measurements

Adam C. Keith, Charles H. Baldwin, Scott Glancy, and E. Knill

Phys. Rev. A 98, 042318 – Published 12 October 2018

Abstract

Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semidefinite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.

Received 16 April 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042318

Physics Subject Headings (PhySH)

Quantum benchmarking Quantum information with trapped ions Quantum tomography

Quantum Information, Science & Technology

Authors & Affiliations

Adam C. Keith^1,2,*, Charles H. Baldwin¹, Scott Glancy^1,†, and E. Knill^1,3

¹Applied and Computational Mathematics Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA
²Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
³Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

^*adam.keith@colorado.edu
^†sglancy@nist.gov

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Schematic describing the measurement model for a single engineered measurement ${F_{i, b}}_{b}$ . A density matrix $ρ$ is rotated by the unitary operator $U_{i}$ , which determines the basis in which $ρ$ is measured. In the measurement, first the underlying POVM ${Π_{k}}_{k}$ is applied, and result $k$ is obtained with probability $P_{k, i}$ . Then, a Markov process acts to give random outcome $b$ with probability $Q_{k, b}$ . In the experiment we only perceive the outcome $b$ . Ellipses and dotted arrows show that there may be an arbitrary finite number of underlying measurement operators $Π_{k}$ and values of $b$ .
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Figure 2
Schematic showing measurement and estimation procedure. Reference experiments begin with the preparation of the known state $ρ_{0}$ , and probe experiments begin with unknown states from $σ = {σ_{j} | j = 1, \dots, s}$ . During each experiment the states are transformed by a process in the set $U = {U_{i} | i = 0 \dots r}$ and then measured with the POVM $F = {F_{b} | b = 1, \dots, M}$ , giving outcome $b$ . For simplicity, we assume every known process acts on each unknown state, but this is not necessary. After $n$ trials of each experiment, histograms ${H_{b, i}^{(j)}}_{b, i}$ for each state are recorded. These histograms are then coarse grained to produce new histograms ${H_{c, i}^{(j)}}_{c, i}$ based on a training data set composed of 10% of the trials randomly selected without replacement from each reference experiment. From the reference experiments' data, we calculate an initial estimate for the POVM ${\hat{F}}^{ini}$ . We divide the likelihood maximization into two concave subproblems: maximization of $L_{1}$ with respect to $σ$ and maximization of $L_{2}$ with respect to $F$ . We alternate between using the $R ρ R$ algorithm to maximize $L_{1}$ and a standard nonlinear optimizer to maximize $L_{2}$ . At each iteration we find maximizing parameters ${\hat{σ}}^{cur}$ and ${\hat{F}}^{cur}$ , and we check a stopping criterion. When the stopping criterion signals that we can stop iterations, we have the numerical maximum likelihood estimates $\hat{σ}$ and $\hat{F}$ .
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Figure 3
Histograms from simulated measurements of two ions. An individual ion fluoresces if it is in the $| ↑ 〉$ (bright) state. We model the distributions of photon counts as Poissonians. In a real experiment the distributions differ from Poissonians due to processes such as repumping from the dark to the bright state. For two ions, there are three possible count distributions corresponding to the total number of ions in the bright state: zero ions bright (mean 2, due to dark counts), one ion bright (mean 20), and two ions bright (mean 40). For one ion bright, the photon detector cannot determine which of the two ions is fluorescing. (a) Relative frequency histograms for both ions in the dark state (blue), one ion on the dark state (orange), and both ions in the bright state (red). (b) Relative frequency histogram for the reference experiment with $U_{1} = U {(\frac{π}{2}, 0)}^{\otimes 2}$ . (c) Relative frequency histogram for the probing experiment with $U_{3} = U {(\frac{π}{2}, \frac{π}{2})}^{\otimes 2}$ .
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Figure 4
Conceptual schematic of bounding expectation values. The likelihood maximization returns an estimate $\hat{σ}$ that is an element of the set of density matrices that maximize the likelihood (green line). Projecting from the set of maximum likelihood density matrices onto the line representing observable $O_{1}$ gives a unique expectation value. Because $O_{1}$ is in the span of the estimated measurement operators, each maximum likelihood density matrix has the same expectation value for $O_{1}$ . Observable $O_{2}$ is only partially contained within the span of estimated measurement operators. The SDP provides upper and lower bounds of its expectation value.
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Figure 5
Illustration of the binning procedure. (a) Relative frequency histogram for the second reference experiment, $U_{1} = U {(\frac{π}{2}, 0)}^{\otimes 2}$ . (b) Same histogram data but after binning. The black lines show the bin boundaries found with the heuristic algorithm.
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