A linear response framework for simulating bosonic and fermionic correlation functions illustrated on quantum computers

Speaker: Efekan Kökcü

​

Department of Physics at

North Carolina State University

​

​

arXiv:2302.10219

​

Kemper Lab

2

Collaborations with:

​

• Jim Freericks (Georgetown)

​

• Bert de Jong, Katie Klymko, Daan Camps, Roel van Beeumen, Akhil Francis (LBNL)

​

• Thomas Steckmann (UMD)

3

Introduction

• Quantum computers are natural fit for simulating spin/fermion models

​

• Response functions: from experiments and simulations

​

​

• In position-momentum or time-frequency basis

​

Phys. Rev. X 8, 041009 (2018)

10.1103/PhysRevB.101.014411

4

Quantum Algorithm(s) for χ

In terms of frequency:

​

5

Quantum Algorithm(s) for χ

In terms of frequency:

​

6

Quantum Algorithm(s) for χ

In terms of frequency:

​

In terms of time:

​

7

Hadamard Test

• Requires an ancilla qubit to be coupled to the system
• If A and B are not local in space, CNOT count increases

8

Linear Response Method

Simulating an experiment is a more natural choice which leads to an ancilla free method to get response functions.

9

Bosonic (commutator) response functions:

=

10

• Momentum selectivity: we can apply any linear combination over qubits since

Momentum selectivity

Our method

Hadamard Test

11

• Fermionic Green’s functions are given as

​

​

​

​

• This requires us to be able to measure the following

​

• We propose two different ways to achieve this within the linear response formalism: 1) Auxiliary operator 2) Post selection

Fermionic (anti-commutator) response functions

12

, with

​

• If satisfies

​

• The method relies on the following identity:

​

Auxiliary operator method:

13

• Finally, if satisfies , we have , which leads to

​

​

​

​

​

• This allows us to measure anti-commutators via the previous method

Auxiliary operator method:

14

• If we choose and we obtain

​

​

• The operator satisfies
• For systems that preserves particle number parity:
• If state has definite parity (even or odd particle number), then

with s = +1 or -1

• This works both for particle conserving and superconducting systems

​

Example: retarded Green’s function

15

• Assume that the Hamiltonian is particle conserving, and the ground state has a definite number of particles (which we denote with N).

Post selection method:

16

Post selection method: (particle conserving only)

17

Post selection method:

18

Su-Schrieffer-Heeger model

• Bosonic correlation function: polarizability (density-density correlation)
• Fermionic correlation function: single particle Green’s function

19

Circuit for SSH Model Response Functions

• Circuit for SSH model to calculate Green’s function on k basis
• We used Algebraic Compression to compress the time evolution circuit

​

​

​

20

Polarizability

Wiggle potential

on site 0

Measure density

on all sites

21

Single particle Green’s function

Noisy simulator results for 8 site SSH model with 1%(10%) noise on 1(2) qubit gates

22

Single particle Green’s function

Results from ibm_auckland for 8 site SSH model with a 14 CNOT circuit with momentum selectivity:

23

• Ancilla free

​

• Momentum and frequency selectivity

​

• Both bosonic and fermionic correlators

​

• More noise robust compared to existing methods

​

​

​

Conclusion

24

Collaborations with:

​

• Jim Freericks (Georgetown)

​

• Bert de Jong, Katie Klymko, Daan Camps, Roel van Beeumen, Akhil Francis (LBNL)

​

• Thomas Steckmann (UMD)