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The Journal of Chemical Physics

Experimental demonstration of simplified quantum process tomography

Ze Wu, S. Li, Wenqiang Zheng, Xinhua Peng, Mang Feng

Published Jan 14, 2013• 11 citations
Publisher

Abstract

The essential tool to characterize dynamics of an open quantum system is quantum process tomography (QPT). Although standard QPT methods are hard to be scalable, simplified QPT approach is available if we have the prior knowledge that the system Hamiltonian commutes with the system-environment interaction Hamiltonian. Using a nuclear magnetic resonance (NMR) quantum simulator, we experimentally simulate dephasing channels to demonstrate the simplified QPT as well as the standard QPT method as a comparison. The experimental results agree well with our predictions which confirm the validity and better efficiency of the simplified QPT.

DOI
10.1063/1.4774119

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The essential tool to characterize dynamics of an open quantum system is quantum process tomography (QPT). Although standard QPT methods are hard to be scalable, simplified QPT approach is available if we have the prior knowledge that the system Hamiltonian commutes with the system-environment interaction Hamiltonian. Using a nuclear magnetic resonance (NMR) quantum simulator, we experimentally simulate dephasing channels to demonstrate the simplified QPT as well as the standard QPT method as a comparison. The experimental results agree well with our predictions which confirm the validity and better efficiency of the simplified QPT.
Developments of quantum techniques have drastically improved the control ability of current quantum processors. Characterizing quantum simulations performed on these processors has become a vital topic in quantum information processing. Traditional characterizing techniques such as quantum process and state tomography fully characterize quantum simulations, but always with the exponentially increased resource cost, unfortunately. It may place their applications only in small-scale quantum systems. Meanwhile, many potential characterizing methods have been put forward in the past decades. In this work we experimentally test and compare different methods for characterizing quantum simulations of Ising Hamiltonians on the nuclear magnetic resonance (NMR) platform, including the recent quantum Hamiltonian tomography with polynomial complexity to estimate the generator of quantum simulations and nonrepetitive direct fidelity estimation with the Monte Carlo method to estimate the fidelity of dynamical states. We also experimentally estimate the parameters of Hamiltonian via full quantum process tomography and quantum state tomography for the comparison. These methods have merits and demerits in accuracy, the required resource, and prior knowledge. Our experiments test the performance of different methods while providing a beneficial reference for characterizing quantum simulations in the future.
We propose and evaluate experimentally an approach to quantum process tomography that completely removes the scaling problem plaguing the standard approach. The key to this simplification is the incorporation of prior knowledge of the class of physical interactions involved in generating the dynamics, which reduces the problem to one of parameter estimation. This allows part of the problem to be tackled using efficient convex methods, which, when coupled with a constraint on some parameters, allows globally optimal estimates for the Krauss operators to be determined from experimental data. Parameterizing the maps provides further advantages: it allows the incorporation of mixed states of the environment as well as some initial correlation between the system and environment, both of which are common physical situations following excitation of the system away from thermal equilibrium. Although the approach is not universal, in cases where it is valid it returns a complete set of positive maps for the dynamical evolution of a quantum system at all times.
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.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.
Quantum process tomography is a procedure by which the unknown dynamical evolution of an open quantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure can be implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelity of a controlled-NOT logic gate and to experimentally investigate the error model for our computer. Based on the latter analysis, we test an important assumption underlying many models of quantum error correction, the independence of errors on different qubits.
We experimentally implement the Sz.-Nagy dilation algorithm to simulate open quantum dynamics on a nuclear magnetic resonance quantum processor. The Sz.-Nagy algorithm enables the simulation of the dynamics of an nnn-qubit system using n+1n+1n+1 qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin-12\frac{1}{2}21​ particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.