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Physical Review A

Quantum simulation using fidelity-profile optimization

Manu Veliparambil Subrahmanian, Anil Kumar

Published May 29, 2014• 12 citations
Publisher

Abstract

Experimental quantum simulation of a Hamiltonian $H$ requires unitary operator decomposition (UOD) of its evolution unitary $U=\text{exp}(\ensuremath{-}iHt)$ in terms of native unitary operators of the experimental system. Here, using a genetic algorithm, we numerically evaluate the most generic UOD (valid over a continuous range of Hamiltonian parameters) of the unitary operator $U,$ termed fidelity-profile optimization. The optimization is obtained by systematically evaluating the functional dependence of experimental unitary operators (such as single-qubit rotations and time-evolution unitaries of the system interactions) to the Hamiltonian ($H$) parameters. Using this technique, we have solved the experimental unitary decomposition of a controlled-phase gate (for any phase value), the evolution unitary of the Heisenberg XY interaction, and simulation of the Dzyaloshinskii-Moriya (DM) interaction in the presence of the Heisenberg XY interaction. Using these decompositions, we studied the entanglement dynamics of a Bell state in the DM interaction and experimentally verified the entanglement preservation procedure of Hou et al. [Ann. Phys. (N.Y.) 327, 292 (2012)] in a nuclear magnetic resonance quantum information processor.

DOI
10.1103/physreva.89.052331

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Experimental quantum simulation of a Hamiltonian HHH requires unitary operator decomposition (UOD) of its evolution unitary U=exp(\ensuremath−iHt)U=\text{exp}(\ensuremath{-}iHt)U=exp(\ensuremath−iHt) in terms of native unitary operators of the experimental system. Here, using a genetic algorithm, we numerically evaluate the most generic UOD (valid over a continuous range of Hamiltonian parameters) of the unitary operator U,U,U, termed fidelity-profile optimization. The optimization is obtained by systematically evaluating the functional dependence of experimental unitary operators (such as single-qubit rotations and time-evolution unitaries of the system interactions) to the Hamiltonian (HHH) parameters. Using this technique, we have solved the experimental unitary decomposition of a controlled-phase gate (for any phase value), the evolution unitary of the Heisenberg XY interaction, and simulation of the Dzyaloshinskii-Moriya (DM) interaction in the presence of the Heisenberg XY interaction. Using these decompositions, we studied the entanglement dynamics of a Bell state in the DM interaction and experimentally verified the entanglement preservation procedure of Hou et al. [Ann. Phys. (N.Y.) 327, 292 (2012)] in a nuclear magnetic resonance quantum information processor.
Quantum simulation of a Hamiltonian H requires unitary operator decomposition (UOD) of its evolution operator, (U=exp(−iHt)U=exp(-i H t)U=exp(−iHt)) in terms of experimentally preferable unitaries. Here, using Genetic Algorithm optimization, we numerically evaluate the most generic UOD for the Hamiltonian, DM interaction in the presence of Heisenberg XY interaction, HDHH_{DH}HDH​. Using these decompositions, we studied the entanglement dynamics of Bell state in the Hamiltonian HDHH_{DH}HDH​ and verified the entanglement preservation procedure by Hou et al. [Annals of Physics 327, 292 (2012)].
Genetic algorithms (GA) are stochastic global search methods based on the mechanics of natural biological evolution, proposed by John Holland in 1975. Here in this thesis, we have exploited possible utilities of Genetic Algorithm optimization in Nuclear Magnetic Resonance (NMR) experiments. We have performed (i ) Pulse sequence generation and optimization for NMR Quantum Information Processing, (ii ) efficient creation of NOON states, (iii ) Composite operator design and (iv ) delay optimization for refocused quantitative INEPT. We have generated time optimal as well as robust pulse sequences for popular quantum gates. A Matlab package is developed for basic Target unitary operator to pulse sequence optimization and is explained with an example. Chapter 1 contains a brief introduction to NMR, Quantum computation and Genetic algorithm optimization. Experimental unitary operator decomposition using Genetic Algorithm is explained in Chapter 2. Starting from a two spin homonu- clear system (5-Bromofuroic acid), we have generated hard pulse sequences for performing (i ) single qubit rotation, (ii ) controlled NOT gates and (iii ) pseudo pure state creation, which demonstrates universal quantum computation in such systems. The total length of the pulse sequence for the single qubit rotation of an angle π/2 is less than 500µs, whereas the conventional method (using a selective soft pulse) would need a 2ms shaped pulse. This substantial shortening in time can lead to a significant advantage in quantum circuits. We also demonstrate the creation of Long Lived Singlet State and other Bell states, directly from thermal equilibrium state, with the shortest known pulse sequence. All the pulse sequences generated here are generic i.e., independent of the system and the spectrometer. We further generalized this unitary operator decomposition technique for a variable operators termed as Fidelity Profile Optimization (FPO) (Chapter 3) and performed quantum simulations of Hamiltonian such as Heisenberg XY interaction and Dzyaloshinskii-Moriya interaction. Exact phase (φ) dependent experimental unitary decompositions of Controlled-φ and Controlled Controlled-φ are solved using first order FPO. Unitary operator decomposition for experimental quantum simulation of Dzyaloshinskii-Moriya interaction in the presence of Heisenberg XY interaction is solved using second order FPO for any relative strengths of interactions (γ) and evolution time (τ ). Experimental gate time for this decomposition is invariant under γ or τ , which can be used for relaxation independent studies of the system dynamics. Using these decompositions, we have experimentally verified the entanglement preservation mechanism suggested by Hou et al. [Annals of Physics, 327 292 (2012)]. NOON state or Schrodinger cat state is a maximally entangled N qubit state with superposition of all individual qubits being at |0 and being at |1 . NOON states have received much attention recently for their high precession phase measurements, which enables the design of high sensitivity…
Precise experimental implementation of unitary operators is one of the most important tasks for quantum information processing. Numerical optimization techniques are widely used to find optimized control fields to realize a desired unitary operator. However, finding high-fidelity control pulses to realize an arbitrary unitary operator in larger spin systems is still a difficult task. In this work, we demonstrate that a combination of the grape algorithm, which is a numerical pulse optimization technique, and a unitary operator decomposition algorithm [Ajoy et al., Phys. Rev. A 85, 030303 (2012)] can realize unitary operators with high experimental fidelity. This is illustrated by simulating the mirror-inversion propagator of an XY spin chain in a five-spin dipolar coupled nuclear spin system. Further, this simulation has been used to demonstrate the transfer of entangled states from one end of the spin chain to the other end.
Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.
Quantum simulation is a promising near term application for quantum information processors with the potential to solve computationally intractable problems using just a few dozen interacting qubits. A range of experimental platforms have recently demonstrated the basic functionality of quantum simulation applied to quantum magnetism, quantum phase transitions and relativistic quantum mechanics. However, in all cases, the physics of the underlying hardware restricts the achievable inter-particle interactions and forms a serious constraint on the versatility of the simulators. To broaden the scope of these analog devices, we develop a suite of pulse sequences that permit a user to efficiently realize average Hamiltonians that are beyond the native interactions of the system. Specifically, this approach permits the generation of all symmetrically coupled translation-invariant two-body Hamiltonians with homogeneous on-site terms, a class which includes all spin- XYZ chains, but generalized to include long-range couplings. Our work builds on previous work proving that universal simulation is possible using both entangling gates and single-qubit unitaries. We show that determining the appropriate 'program' of unitary pulse sequences which implements an arbitrary Hamiltonian transformation can be formulated as a linear program over functions defined by those pulse sequences, running in polynomial time and scaling efficiently in hardware resources. Our analysis extends from circuit model quantum information to adiabatic quantum evolutions, representing an important and broad-based success in applying functional analysis to the field of quantum information.
Abstract. Quantum simulation is a promising near term application for quantum information processors with the potential to solve computationally intractable problems using just a few dozen interacting qubits [1, 2, 3]. A range of experimental platforms have recently demonstrated the basic functionality of quantum simulation applied to quantum magnetism, quantum phase transitions, and relativistic quantum mechanics [4, 5, 6, 7, 8, 9, 10, 11]. However, in all cases, the physics of the underlying hardware restricts the achievable inter-particle interactions and forms a serious constraint on the versatility of the simulators. To broaden the scope of these analog devices, we develop a suite of pulse sequences that permit a user to efficiently realize average Hamiltonians that are beyond the native interactions of the system [12, 13]. Specifically, this approach permits the generation of all symmetrically coupled translation-invariant two-body Hamiltonians with homogeneous on-site terms, a class which includes all spin-1/2 XYZ chains, but generalized to include long-range couplings. Our work builds on previous work proving that universal simulation is possible using both entangling gates and single-qubit unitaries [14, 15, 16]. We show that determining the appropriate “program ” of unitary pulse sequences which implements an arbitrary Hamiltonian transformation can be formulated as a linear program over functions defined by those pulse sequences, running in polynomial time and scaling efficiently in hardware resources. Our analysis extends from circuit model quantum information to adiabatic quantum evolutions, representing an important and broad-based success in applying functional analysis to the field of quantum information. ar X iv