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arXiv:2201.07687 · Physical Review A

Simulating open quantum dynamics on an NMR quantum processor using the Sz.-Nagy dilation algorithm

Akshay Vijay Gaikwad, Arvind, Kavita Dorai

Published Aug 22, 2022• 13 citations
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Abstract

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 $n$-qubit system using $n+1$ 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-$\frac{1}{2}$ 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.

DOI
10.1103/physreva.106.022424

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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.
We propose and experimentally demonstrate an efficient framework for the quantum simulation of quantum channels in nuclear magnetic resonance (NMR). Our approach relies on the suitable decomposition of nonunitary operators in a linear combination of ddd unitary ones, which can be then experimentally implemented with the assistance of a number of ancillary qubits that grows logarithmically in ddd. As a proof-of-principle demonstration, we realize the quantum simulation of three quantum channels for a single-qubit: phase damping, amplitude damping, and depolarizing channels. For these paradigmatic cases, we measure key features, such as the fidelity of the initial state and the associated von Neumann entropy for a qubit evolving through these channels. Our experiments are carried out using nuclear spins in a liquid sample and NMR control techniques.
NMR quantum computing with qubit systems represented by nuclear spins (I=12) in small molecules in liquids has led to the most successful experimental quantum information processors so far. We use the quadrupolar spin-32 sodium nuclei of a NaNO3 single crystal as a virtual two-qubit system. The large quadrupolar coupling in comparison with the environmental interactions and the usage of strongly modulating pulses allow us to manipulate the system fast enough and at the same time keeping the decoherence reasonably slow. The experimental challenge is to characterize the "calculation" behavior of the quantum processor by process tomography which is here adapted to the quadrupolar spin system. The results of a selection of quantum gates and algorithms are presented as well as a detailed analysis of experimental results.
This thesis actively focuses on designing, analyzing, and experimentally implementing various QST and QPT protocols using an NMR ensemble quantum processor and superconducting qubit-based IBM cloud quantum processor. Part of the thesis also includes a study of duality quantum simulation algorithms and Sz-Nagy's dilation algorithm on NMR where several 2-qubit non-unitary quantum channels were simulated using only a single ancilla qubit. The work carried out in the thesis mainly addresses several important issues in experimental QST and QPT which include: i) dealing with invalid experimental density (process) matrices using constraint convex optimization (CCO) method, ii) scalable QST and QPT using incomplete measurements via compressed sensing (CS) algorithm and artificial neural network (ANN) technique, iii) selective and direct measurement of unknown quantum states and processes using the concept of quantum 2-design states and weak measurement (WM) approach and iv) quantum simulation and characterization of open quantum dynamics using the dilation technique.
Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.
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.
In the present work the first virtual 2 Qubit solid state NMR quantum processor, using a spin- 3/2 Na quadrupolar nucleus in a NaNO3 single crystal, is presented. Methods from liquid NMR quantum information processing, like strongly modulating pulses, are adopted to the quadrupolar spin system to implement the unitary operations and to generate pseudo pure states. Experimentally a set of quantum algorithms as well as the decoherence process is characterized by quantum process tomography which is developed for the quadrupolar spin system. A detailed analyses of the basic conditions of the experimental quantum process tomography procedure is presented, showing that even in such non-Markovian systems this method is valid.