Quantum state tomography and purity measurement of a multiqubit system with random single-pulse rotations

Yariv Yanay and Charles Tahan

Phys. Rev. A 103, 062411 – Published 11 June 2021

Abstract

Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as $\sum_{a, b} {| Δ ρ_{a b} |}^{2} \sim 5^{N} / N_{tot}$ for full tomography, and ${Δ μ}^{2} \sim 7^{N} / N_{tot}^{2}$ for purity estimation, for $N$ qubits and $N_{tot}$ measurements performed.

Received 21 January 2021
Accepted 24 May 2021

DOI:https://doi.org/10.1103/PhysRevA.103.062411

Physics Subject Headings (PhySH)

Entanglement measures Quantum entanglement Quantum simulation Quantum tomography

Quantum Information, Science & Technology

Authors & Affiliations

Yariv Yanay ^* and Charles Tahan

Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, Maryland 20740, USA

^*yariv@lps.umd.edu

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
Quantum state tomography (QST) via random rotations. From top to bottom: An $N$ -qubit system is repeatedly prepared in some state $\hat{ρ}$ ; we apply each of the $N_{U}$ unitary rotations, randomly sampled from some distribution; for each unitary, we perform $N_{M}$ measurements of the state of the system, measuring all qubits in the laboratory $Z$ axis; from all of this we deduce the probability map $P (s | {\hat{U}}_{i})$ [see Eq. (10)]. We then apply an inverse transformation $M^{- 1}$ [see Eq. (11)] and average over all rotations to obtain the density matrix.
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Figure 2
Scaling of the measurement variance in full QST, including statistical variance depending on the number of unitaries (left) and number of measurements (right) [see Eq. (14)]. They are shown for 3, 4, and 5 qubits, at the optimal point $g_{q} = \sqrt{2} ν_{q}$ , for 60 different density matrices each. We generate these density matrices in three different ways, as described in Appendix pp1. The dotted lines show the amortized estimates given by Eq. (37c), ${(Δ_{tom}^{U})}_{amo}^{2} \approx {(5 / 2)}^{N} μ$ on the left; and the calculated value of Eq. (38), ${(Δ_{tom}^{M})}^{2} = 5^{N}$ .
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Figure 3
Scaling of the statistical variance in purity measurement with the number of qubits in the system [see Eq. (26)], comparing numerical results and analytical estimates at the optimal working point, $g_{q} = \sqrt{2} ν_{q}$ for all qubits. In each plot, we show the results from each of 60 different density matrices randomly generated in three different ways, as described in Appendix pp1. The dotted lines correspond to the amortized estimates given in Table 2 at the optimal working point. We observe good agreement between our predictions and the numerical calculations.
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Figure 4
The smallest linear combination of frequencies, $θ_{a - b - i + j}$ , as defined in Eq. (42), for an exponential frequency ladder $λ_{q} = λ_{0} / 3^{q}, q = 0, ..., N - 1$ (blue circles) and for frequencies randomly chosen from a uniform distribution $λ_{q} \in [0, λ_{0}]$ (yellow squares). For random frequencies, we average over 200 randomizations and show the logarithmic mean and standard deviation. The smallest linear combination decreases with qubit number as $min [θ] \sim 1 / 3^{N}, 1 / 4 . 6^{N}$ , respectively, implying that the maximum rotation time $T$ must increase exponentially with qubit number. We find that variations on these arrangements such as a uniform offset for all frequencies or nonuniform sampling of frequencies (not shown here) do not significantly change the scaling behavior.
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Figure 5
Verification of our results via numerical experiments. Here, the shaded area in each curve corresponds to the standard deviation of 15 numerical experiments where the density matrix is estimated as per the protocol of Sec. 3, while the solid curve is the analytical expression derived in the main text. This is done at four different target purities, and for $N = 3, 4, 5$ qubits. The top, middle, and bottom curve for each $N$ corresponds to measuring each unitary $N_{M} = 10$ , 20, and 40 times, respectively. We observe remarkable agreement.
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Figure 6
Comparison between the calculated statistical deviation of the measured density matrix from the real density matrix $Δ_{tom}^{2}$ , with their trace distance, in the numerical experiments described in Fig. 5. Shown for $N = 3, 4, 5$ and $N_{M} = 10, 20, 40$ measurements per unitary. Each point represents the measured density matrix after $N_{U}$ rotations, and the solid lines show the bound on the ratio, $2^{N}$ .
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