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arXiv:0711.1017 · Journal of Physics A Mathematical and Theoretical

Optimizing quantum process tomography with unitary 2 -designs

A J Scott

Published Jan 23, 2008• 119 citations
arXivHTMLPDF

Abstract

We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.

DOI
10.1088/1751-8113/41/5/055308

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We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.
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.
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.
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.
We consider the problem of estimating an SU(d) quantum operation when n copies of it are available at the same time. It is well known that, if one uses a separable state as the input for the unitaries, the optimal mean square error will decrease as 1/n. However it is shown here that, if a proper entangled state is used, the optimal mean square error will decrease at a 1/n^2 rate. It is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures) can be used to design optimal input states. Although 2-designs are believed to exist for every dimension, this has not yet been proven. Therefore, we give an alternative input state based on approximate 2-designs which can be made arbitrarily close to optimal. It is shown that measurement strategies which are based on local operations and classical communication between the ancilla and the rest of the system can be optimal.
We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on Abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.