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:
Calculations that have sign problems:
Closely related, but mild: determining the mass of a proton
Calculations without sign problems:
Set up an analogy between the quantum computer and the system to be simulated, and treat the computer like a (perfectly controlled) laboratory.
This is a spin from QM.
What's the Hilbert space?
In other words, superpositions of all possible bitstrings.
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.
In principle, we measure any Hermitian operator. In practice, we measure
Each measurement yields
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.
Current best:
Each qubit can undergo
Are large processors worth it?
This restricts the set of possible operations, as well. We only have permutations:
Example: two-bit adder
In general, given a classical circuit for
Assume we can evolve under
So, time-evolve by rapidly alternating between two Hamiltonians
Each lattice site has a degree of freedom with Hilbert space
Some Hamiltonian
When correlations are large, the lattice structure is irrelevant, hence "field theory".
The Hamiltonian of a free particle moving on
Hilbert space is
We can work with momentum eigenstates instead.
The Hamiltonian is gauge-invariant.
Here's a projection operator:
Four states, naively:
Gauge transformation takes
We need a mapping between the Hilbert spaces. Locality is nice.
This induces a "nice" map
Implement
One link only
Diagonal in Fourier space
Mutually commuting terms
Four links
Diagonal (in our basis)
Mutually commuting terms
We'd like to simulate
After adding an extra term to the action...
Construct a dimensionless quantity from: Wilson flow, critical temperature
Particle masses not yet measured...
The physical Hilbert space is not
But, the Hilbert space on the quantum computer is isomorphic to that space!
If we start in a gauge-invariant state, we stay gauge-invariant.
The hadronic tensor captures nonperturbative (in QCD coupling) information about the proton. For electron-proton scattering, to leading order in
Naive state preparation: couple to a heat bath and cool the system. Expensive!
Alternative: adiabatic state preparation
Other proposals:
arXiv:1709.08250)1908.07051)Hard to analyze, impossible to test
Take a time-varying (slowly) Hamiltonian
Prepare an eigenstate of
When
Time needed to prepare ground state:
Restrict to a certain sector of Hilbert space:
How do we measure
Linear response: differentiate with respect to
Shear/bulk viscosity, conductivity, etc...
The expectation value still has the form of linear response:
State preparation dominates the gate cost of the calculation.
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.
Boltzmann factor
Partition function
Expectation values
Density matrix
Partition function
Expectation values
Several ways of turning
Equality holds when
We want
Therefore: sample pairs of states
How to compute
Preparing a basis state
Even computing
For small
(Periodic boundary conditions)
(Open boundary conditions)
One step of time evolution requires
With state prep, need to evolve for
Characteristically,
Without state prep, each quantum computation is now
At the cost of needing to perform many more computations (sampling
The QC calculation is shorter, but we need many of them.