Selective and efficient quantum process tomography

Abstract

In this paper we describe in detail and generalize a method for quantum process tomography that was presented in [1]. The method enables the efficient estimation of any element of the χ-matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required to implement it scale polynomically with the number of qubits of the system. The estimation of all diagonal elements of the χ-matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, that is based on efficient estimation by random sampling over a set of states forming a 2-design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.

Quantum pseudorandomness from cluster-state quantum computation

“…Remarkably, random pure states saturate the classical communication capacity of a noisy quantum channel [9], and allow superdense coding of arbitrary quantum states [10]. Random unitaries find applications in tasks ranging from approximate encryption and remote state preparation [11] to unbiased noise estimation [12,13] and selective process tomography [14].However, implementing exact randomization on a quantum computer is inefficient, as the number of required elementary gates grows exponentially with the number of qubits. Still, it has been shown [5,12,15] that one can generate pseudo-random (PR) quantum states and unitary operators which satisfy certain practical tests of randomness using only a polynomial number of gates.…”

Selective and efficient quantum state tomography and its application to quantum process tomography

“…Finally, we showed that this protocol shares some properties with SEQPT, a method presented in [13,14]. The main difference is that the use of ancillas is a way to avoid the preparation of the special states of the 2-design and sampling on them.…”

“…The main idea there is to follow the proceedure described by the circuit shown in Figure 4, and to estimate the average answer averaging over the entire Hilbert space of the system using the Haar measure for that purpose. As it is shown in [13,14], that average cn be directly related to the matrix element χ ab as…”

“…For this to be possible, we need that any state from the basis B can be efficently prepared in a controlled manner. This method, when applied for implementing QPT results in a protocol for efficient and selective QPT that is equivalent to the one presented in [13,14], illustrating one of the virtues of such a selective and efficient QST scheme. This paper if organized as follows.…”

“…However, there are some quantum algorithms that allow to efficiently extract important information about a given quantum process [10][11][12][13][14][15][16]. These algorithms do not require performing QST on the final states but measuring quantities such as survival probabilities (or transition probabilities).…”