Scalable reconstruction of unitary processes and Hamiltonians

M. Holzäpfel, T. Baumgratz, M. Cramer, and M. B. Plenio

Phys. Rev. A 91, 042129 – Published 29 April 2015

Abstract

Based on recently introduced efficient quantum state tomography schemes, we propose a scalable method for the tomography of unitary processes and the reconstruction of one-dimensional local Hamiltonians. As opposed to the exponential scaling with the number of subsystems of standard quantum process tomography, the method relies only on measurements of linearly many local observables and either (a) the ability to prepare eigenstates of locally informationally complete operators or (b) access to an ancilla of the same size as the to-be-characterized system and the ability to prepare a maximally entangled state on the combined system. As such, the method requires at most linearly many states to be prepared and linearly many observables to be measured. The quality of the reconstruction can be quantified with the same experimental resources that are required to obtain the reconstruction in the first place. Our numerical simulations of several quantum circuits and local Hamiltonians suggest a polynomial scaling of the total number of measurements and post-processing resources.

Received 8 December 2014
Revised 28 March 2015

DOI:https://doi.org/10.1103/PhysRevA.91.042129

Authors & Affiliations

M. Holzäpfel¹, T. Baumgratz^1,2, M. Cramer¹, and M. B. Plenio¹

¹Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
²Clarendon Laboratory, Department of Physics, University of Oxford, OX1 3PU Oxford, United Kingdom

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Vol. 91, Iss. 4 — April 2015

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Figure 1
Two equivalent ways to obtain the necessary measurements for the scalable process tomography scheme. Top row: System and ancilla are prepared in a maximally entangled state $| Φ 〉 \propto ⨂_{i = 1}^{N} {[| 0 〉}_{2 i - 1} {| 0 〉}_{2 i} + {| 1 〉}_{2 i - 1} {| 1 〉}_{2 i}]$ (with pairwise entanglement indicated by vertical lines), the channel $E$ is applied on the system, and products of Pauli observables are measured on all blocks of $r$ consecutive sites (exemplified for $r = 3$ and ${\hat{σ}}_{2}^{y} \otimes {\hat{σ}}_{3}^{x} \otimes {\hat{σ}}_{4}^{z}$ ), which serves as input to reconstruct the resulting state ${\hat{ϱ}}_{E}$ and thus the channel $E$ . Bottom row: Equivalently, these expectation values may be obtained without the ancillary system. The initial states have eigenstates of Pauli matrices (here ${\hat{P}}_{↑_{x}}, {\hat{P}}_{↓_{x}}$ , so those of ${\hat{σ}}_{2}^{x}$ ) on those sites where a Pauli matrix would have been measured on the ancilla. After application of the channel, the system part of the ancilla + system measurement is carried out (here that of ${\hat{σ}}_{1}^{y} \otimes {\hat{σ}}_{2}^{z}$ ).
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Figure 2
Reconstruction of the quantum circuits in Eq. (10). Figures show the fidelity in Eq. (8) of the exact unitary and its tomographic reconstruction as a function of the system size $n$ for $M = \infty$ (left) and as a function of the number $M$ of measurements per local observable for different $n$ (right). Reconstruction uses measurements on all blocks of $r$ consecutive qubits only. Lines are guides to the eye. The numerical data suggest a linear scaling of ${(1 - \sqrt{F})}^{1 / 2}$ with system size $n$ and $1 / \sqrt{M}$ .
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Figure 3
Left: Fidelity between exact state $| ψ_{E} 〉$ and tomographic estimate $| ψ_{E}^{rec} 〉$ of a unitary time evolution $\hat{U} = e^{- i \hat{H} t}$ , with $\hat{H}$ the isotropic Heisenberg Hamiltonian on $n$ qubits. The tomographic estimate is based on complete measurements on blocks of $r = 3$ consecutive qubits with $M$ measurements per observable. Right: Inverse relative distance $D$ between $\hat{H}$ and ${\hat{H}}_{rec}$ reconstructed from $| ψ_{E}^{rec} 〉$ . The unit of time is $t_{n} = 1 / ∥ \hat{H} ∥ \sim 1 / n$ , i.e., inversely proportional to the number of qubits, and the gray area indicates where reconstruction is expected to fail due to Eq. (5). The gray lines without any markers show the expected behavior for small times and times near $π$ [see Eq. (11) and the main text]. The data indicates that for $M < \infty, M / n^{2}$ fixed and $t / t_{n}$ fixed and sufficiently small, $D (\hat{H}, {\hat{H}}_{rec})$ is largely independent of $n$ .
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Figure 4
Quality of the reconstruction ${\hat{H}}_{rec}$ for $\hat{H}$ the critical Ising (left), isotropic Heisenberg (middle), and a random nearest-neighbor Hamiltonian (right). ${\hat{H}}_{rec}$ was obtained from $U (t^{'} - t)$ with $t / t_{n} = 3.51$ fixed and $U (t^{'} - t) = U^{†} U^{'}$ was obtained from tomographic estimates of $U = e^{- i \hat{H} t}$ and $U^{'} = e^{- i \hat{H} t^{'}}$ . The inverse relative error $1 / D (\hat{H}, {\hat{H}}_{rec})$ is shown for $H_{rec}$ (diamonds) and for the projection of $H_{rec}$ onto a nearest-neighbor Hamiltonian (circles). $\hat{H}$ acts on $n = 32$ qubits and tomographic estimates are based on complete measurements on blocks of $r = 5$ consecutive qubits with $M$ measurements per observable. The unit of time is $t_{n} = 1 / ∥ \hat{H} ∥ \sim 1 / n$ . Most properties are similar to Fig. 3 right where ${\hat{H}}_{rec}$ was reconstructed from an estimate of $U (t)$ , with the time difference $t^{'} - t$ taking the role of the time $t$ . Lines are guides to the eye.
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Figure 5
Left: Fidelity between exact state $| ψ_{E} 〉$ and tomographic estimate $| ψ_{E}^{rec} 〉$ for a unitary time evolution $\hat{U} = e^{- i \hat{H} t}$ with $\hat{H}$ the Hamiltonian of the critical Ising model (top row) and a Hamiltonian with random nearest-neighbor interaction (bottom row). The Hamiltonians act on $n$ qubits and the tomographic estimate is based on complete measurements on blocks of $r = 3$ consecutive qubits with $M$ measurements per observable. Right: Inverse relative distance between $\hat{H}$ and ${\hat{H}}_{rec}$ reconstructed from $| ψ_{E}^{rec} 〉$ . The unit of time is $t_{n} = 1 / ∥ \hat{H} ∥ \sim 1 / n$ . Results are very similar to the isotropic Heisenberg data shown in Fig. 3 in the main text.
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