Quantum Simulation of Gauge Theories

Scott Lawrence
in collaboration with Andrei Alexandru, Paulo Bedaque, Siddhartha Harmalkar,
Hersh Kumar, Henry Lamm, Neill Warrington, Yukari Yamauchi
"The best model of a cat is another cat"
Norbert Wiener

Neutron stars and little bangs


Thomas McCauley/CERN

The lattice, briefly

QCD is a non-perturbative theory. (There's no small parameter about which to Taylor expand.)

At every link lives a unitary matrix.

The action is a sum over all `plaquettes'.

This is a theory of with a finite number of degrees of freedom. Expectation values are given by:

A sign problem


What calculations have a sign problem?

Calculations that have sign problems:

Closely related, but mild: determining the mass of a proton

Calculations without sign problems:

Overview

The best model of a quantum system is another quantum system.
Norbert Wiener, almost
A quantum computer is a quantum system evolved in real-time.

Set up an analogy between the quantum computer and the system to be simulated, and treat the computer like a (perfectly controlled) laboratory.

From Bit to Qubit

This is a spin from QM.

A Quantum Computer

What's the Hilbert space?

In other words, superpositions of all possible bitstrings.

Not entangled
Entangled

Gates

All operations must be unitary (conservation of probability). On one qubit:

On two qubits, the controlled-not operation:

These fundamental gates are sufficient to construct any unitary we want.

Measurement

In principle, we measure any Hermitian operator. In practice, we measure acting on each qubit.

Each measurement yields or . Probability of is .

Thus, we require many measurements for a precise result. This is "shot noise".

Hermitian operators are exactly those which may appear as terms in the Hamiltonian.

State of the art


Current best: qubits.

Each qubit can undergo operations before decohering.

Are large processors worth it?

What is a classical computer?

A quantum computer constantly being measured
is okay; gets destroyed

This restricts the set of possible operations, as well. We only have permutations:

and not

Classical algorithms are quantum algorithms

Any classical circuit can be re-labelled as a quantum circuit.

Example: two-bit adder

In general, given a classical circuit for , we can obtain

Suzuki-Trotter decomposition

Assume we can evolve under and . How to get evolution under ?

So, time-evolve by rapidly alternating between two Hamiltonians

Field theories, the finite way

Each lattice site has a degree of freedom with Hilbert space . The whole system has Hilbert space

Some Hamiltonian couples the different lattice sites. For a spin system, we might have

When correlations are large, the lattice structure is irrelevant, hence "field theory".

Quantum mechanics on a group

The Hamiltonian of a free particle moving on :

Hilbert space is , the space of complex functions on .

We can work with momentum eigenstates instead.

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Gauge theories

Gauge symmetry


The Hamiltonian is gauge-invariant.

Hilbert space

Only gauge-invariant states are physical.

Here's a projection operator:

Example: gauge theory

Four states, naively: , , , . But these are not gauge-invariant!

Gauge transformation takes and .

Simulating a field theory: general principles

We need a mapping between the Hilbert spaces. Locality is nice.

This induces a "nice" map

Implement via Suzuki-Trotter

Time evolution

Kinetic

One link only

Diagonal in Fourier space

Mutually commuting terms

Potential

Four links

Diagonal (in our basis)

Mutually commuting terms

Example: gauge theory (again)

Valentiner gauge theory

We'd like to simulate , but is infinite-dimensional.

We can approximate by a finite subgroup.

Is this approximation any good?

After adding an extra term to the action...

Construct a dimensionless quantity from: Wilson flow, critical temperature

Particle masses not yet measured...

Maybe

Gauge invariance

The physical Hilbert space is not .

But, the Hilbert space on the quantum computer is isomorphic to that space!

Time evolution is gauge-invariant.

If we start in a gauge-invariant state, we stay gauge-invariant.

The hadronic tensor

The hadronic tensor captures nonperturbative (in QCD coupling) information about the proton. For electron-proton scattering, to leading order in :

Preparing an interesting state

Naive state preparation: couple to a heat bath and cool the system. Expensive!

Alternative: adiabatic state preparation

Other proposals:

Hard to analyze, impossible to test

Adiabatic theorem

Take a time-varying (slowly) Hamiltonian .

Prepare an eigenstate of , with a gap of .

When , time-evolution will keep us in the eigenstate.

Time needed to prepare ground state:

Making a proton

Restrict to a certain sector of Hilbert space:


  • Free fermions and glue (massive)
  • Ground state exactly prepared
  • Small gap ()
  • Hadrons
  • Large gap ()
Total circuit size:

Measuring the hadronic tensor

How do we measure ?

Linear response: differentiate with respect to

Another target: transport coefficients

Shear/bulk viscosity, conductivity, etc...

The expectation value still has the form of linear response:

This algorithm looks a lot like a physical experiment!

Avoiding state preparation

State preparation dominates the gate cost of the calculation.

Can we skip state preparation?

Classical lattice methods are very good at simulating thermodynamics; can't do real-time.

Quantum computers simulate real-time evolution easily; thermodynamics can be expensive.

Hybrid algorithm: thermodynamics classically, evolution quantumly

The Euclidean lattice

Classical thermodynamics

Boltzmann factor

Partition function

Expectation values

Quantum thermodynamics

Density matrix

Partition function

Expectation values


Several ways of turning into a probability distribution to sample. We could take the diagonal

Equality holds when is diagonal. (Common in lattice QCD.)

The real-time oracle

We want ; the operator isn't diagonal!

Therefore: sample pairs of states , distributed by .


How to compute ? This part is done on a quantum computer.

Preparing a basis state is cheap. Start with the states and look at .

How to sample

How do we sample ?

Even computing is hard! Turns out, sampling is easier:

For small , the summand is easily computed. This is a joint distribution over . Marginalized, it approximates .


Diagonal operators

(Periodic boundary conditions)

Arbitrary operators

(Open boundary conditions)

How much do we save?

One step of time evolution requires operations.

With state prep, need to evolve for steps.

Characteristically, . Per measurement:


Without state prep, each quantum computation is now

At the cost of needing to perform many more computations (sampling ).

A big advantage for noisy quantum processors.

The catch

Signal-to-noise problem!

The QC calculation is shorter, but we need many of them.

Probably only useful for near-term, noisy processors.
Thanks for watching!
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