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arXiv:1808.10835

Correlation-assisted process tomography

Matteo Caiaffa, Marco Piani

Published Aug 31, 2018• 0 citations
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Abstract

Standard quantum process tomography on a $d$-dimensional input is performed by preparing several states of an input probe that then evolve under the action of the quantum channel corresponding to the progress. The final states of the probe are reconstructed by means of state tomography. An alternative is offered by ancilla-assisted process tomography: a single probe-ancilla state is used, and the correlations existing between probe and ancilla are exploited to fully reconstruct the information on the channel. In order for ancilla-assisted process tomography to be possible, the probe-ancilla input state does not need to be entangled, but still needs to have maximal operator Schmidt rank. Here we establish and analyze a framework for process tomography that interpolates between these two methods, aiming at exploiting any correlations that may exist between probe and ancilla to allow process tomography with as few input preparations as possible, when the probe-ancilla state may be operator-Schmidt-rank deficient. The main object of our investigation is the minimal number of initial local operations on the input probe for a given starting probe-ancilla state that are needed to allow process tomography. We prove that such a number scales inversely proportional to the operator Schmidt rank of the input probe-ancilla state for arbitrary local input processing. We also provide results for the case where this initial local processing is restricted to be a unitary rotation, in particular showing that the mentioned scaling is satisfied for pure input entangled states, and that in the case of mixed states there might be extreme cases where the fixed input probe-ancilla state provides information on a $\approx d^2/2$-dimensional subspace of the operator space and a single additional input unitary rotation allows process tomography.

DOI
10.48550/arxiv.1808.10835

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Standard quantum process tomography on a ddd-dimensional input is performed by preparing several states of an input probe that then evolve under the action of the quantum channel corresponding to the progress. The final states of the probe are reconstructed by means of state tomography. An alternative is offered by ancilla-assisted process tomography: a single probe-ancilla state is used, and the correlations existing between probe and ancilla are exploited to fully reconstruct the information on the channel. In order for ancilla-assisted process tomography to be possible, the probe-ancilla input state does not need to be entangled, but still needs to have maximal operator Schmidt rank. Here we establish and analyze a framework for process tomography that interpolates between these two methods, aiming at exploiting any correlations that may exist between probe and ancilla to allow process tomography with as few input preparations as possible, when the probe-ancilla state may be operator-Schmidt-rank deficient. The main object of our investigation is the minimal number of initial local operations on the input probe for a given starting probe-ancilla state that are needed to allow process tomography. We prove that such a number scales inversely proportional to the operator Schmidt rank of the input probe-ancilla state for arbitrary local input processing. We also provide results for the case where this initial local processing is restricted to be a unitary rotation, in particular showing that the mentioned scaling is satisfied for pure input entangled states, and that in the case of mixed states there might be extreme cases where the fixed input probe-ancilla state provides information on a ≈d2/2\approx d^2/2≈d2/2-dimensional subspace of the operator space and a single additional input unitary rotation allows process tomography.
Quantum process tomography (QPT) is a fundamental task to characterize the dynamics of quantum systems. In contrast to standard QPT, ancilla-assisted process tomography (AAPT) framework introduces an extra ancilla system such that a single input state is needed. In this paper, we extend the two-stage solution, a method originally designed for standard QPT, to perform AAPT. Our algorithm hasO(MdA2dB2)O(Md_{A}^{2}d_{B}^{2})O(MdA2​dB2​)computational complexity whereMMMis the type number of the measurement operators,dAd_{A}dA​is the dimension of the quantum system of interest, anddBd_{B}dB​is the dimension of the ancilla system. Then we establish an error upper bound and further discuss the optimal design on the input state in AAPT. A numerical example on a phase damping process demonstrates the effectiveness of the optimal design and illustrates the theoretical error analysis.
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.
Characterising multi-time quantum processes is essential for analysing temporally correlated noise and for designing effective control and mitigation strategies. A complete operational description through multi-time process tomography requires an informationally complete set of probes, which necessarily includes non-deterministic intermediate operations. On present-day quantum devices, such operations are commonly implemented using mid-circuit measurements and reset, which are technologically limited and can introduce noise and overhead in terms of ancilla requirement. In this work, we study the minimal ancillary dimension required for complete characterisation of multi-time processes. We show that sequential interactions with a single qubit ancilla can generate an informationally complete family of correlated probes for processes of arbitrary length, without requiring mid-circuit measurements or reset. Our result provides a resource-efficient route for complete multi-time process tomography and establishes that one qubit of coherent ancillary memory suffices for full reconstruction of arbitrary multi-time dynamics.
Lu \textit{et al.} [Ann. Phys. (Berlin) \textbf{534}, 2100550 (2022)] posed the question of whether bound entangled states can be used for ancilla-assisted quantum process tomography (AAQPT). In this work, we answer this question in the affirmative by explicitly demonstrating that certain bound entangled states can be used for AAQPT. We further show that, although local filtering operations may improve the trace norm of the realignment criterion, they are essentially ineffective for AAQPT, as they render the resulting states \emph{unfaithful} and therefore unsuitable for reliable process reconstruction. Further, we investigate the efficiency of bound entangled states for AAQPT and establish quantitative bounds by comparing their performance with Werner and isotropic states. Our results therefore provide a new application of bound entangled states in the context of ancilla-assisted quantum process tomography.
A fundamental task in photonics is to characterise an unknown optical process, defined by properties such as birefringence, spectral response, thickness and flatness. Amongst many ways to achieve this, single-photon probes can be used in a method called quantum process tomography (QPT). Furthermore, QPT is an essential method in determining how a process acts on quantum mechanical states. For example for quantum technology, QPT is used to characterise multi-qubit processors and quantum communication channels; across quantum physics QPT of some form is often the first experimental investigation of a new physical process, as shown in the recent research into coherent transport in biological mechanisms. However, the precision of QPT is limited by the fact that measurements with single-particle probes are subject to unavoidable shot noise---this holds for both single photon and laser probes. In situations where measurement resources are limited, for example, where the process is rapidly changing or the time bandwidth is constrained, it becomes essential to overcome this precision limit. Here we devise and demonstrate a scheme for tomography which exploits non-classical input states and quantum interferences; unlike previous QPT methods our scheme capitalises upon the possibility to use simultaneously multiple photons per mode. The efficiency---quantified by precision per photon used---scales with larger photon number input states. Our demonstration uses four-photon states and our results show a substantial reduction of statistical fluctuations compared to traditional QPT methods---in the ideal case one four-photon probe state yields the same amount of statistical information as twelve single probe photons.
The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\ensuremath{\varepsilon}$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\ensuremath{\chi}$ matrix characterizing the process $\ensuremath{\varepsilon}$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.