Lecture Note
Lattice Gauge Theory on a Noisy Quantum Computer
-
University:
Boston University -
Course:
CAS PY 105 | Physics 1 Academic year:
2022
-
Views:
249
Pages:
42
Author:
Amiah Tran
Lattice Gauge Theory
on a Noisy Quantum Computer What’s the problem?
Quantum systems may not be exactly solved/simulated efficiently with
any existing classical algorithms running on classical computers.
The computational resource and time complexity grow exponentially
with the system size. Interesting problems in lattice gauge theories
and many-body dynamics are required to use huge amount of
computational resource or are incapable of being simulated. Here comes QC
As R.P. Feynman stated in 1982, a computer processing
quantum mechanically should be the most suitable way to
simulate quantum systems .
Any fermionic systems as well as bosonic systems may be
translated into a Pauli-basis system which a digital quantum
computer can process. Then, we only need poly(n)
computational resource for simulation. Quick Recap of QC
● Classical digital bit -> 0 or 1
● Quantum digital bit -> α|0>+β|1>=(α, β)
○ Probability measuring |0> is |α|^2
○ Probability measuring |1> is |β|^2 Quick Recap of QC
Quantum gate representation
-> Matrix (∈SU(2^N)) multiplications on the vector
Example:
Simple quantum circuit.
≅
Input
|ψout> = U2U1|ψin>
Output Quick Recap of QC
Quantum gate representation
-> Matrix (∈SU(2^N)) multiplications on the vector
acts as a rotation in a complex space:
E.g. 1-qubit:
SU(2)/Z2≅O(3)
U What is this project all about?
● Investigating a specific model for lattice gauge
theory (2+1d U(1) theory) on a spin basis. What is this project all about?
● Investigating a specific model for lattice gauge
theory (2+1d U(1) theory) on a spin basis.
● Construct an optimized quantum circuit for it. What is this project all about?
● Investigating a specific model for lattice gauge
theory (2+1d U(1) theory) on a spin basis.
● Construct an optimized quantum circuit for it.
● Run it on the IBM Q machine. What is this project all about?
● Investigating a specific model for lattice gauge
theory (2+1d U(1) theory) on a spin basis.
● Construct an optimized quantum circuit for it.
● Run it on the IBM Q machine.
● Since the device is noisy, it requires some noise
mitigation technique. Hamiltonian Formalism of
2+1D Quantum Link Model Hamiltonian Formalism of
2+1D Quantum Link Model
For QED (U(1)):
Algebra:
Into spin algebra Now we have the spin model
For a single triangle plaquette:
qubit Now we have the spin model
For single triangle:
We are interested in time evolution:
with t/n small enough. Trotterization
Computation of <0|U(t)|0> (exact).
Around t/n ≃ 0.1 for useful simulation. Electric term:
For single triangle:
Circuit
≅ Coupling term:
For single triangle:
Circuit
≅ Plaquette term:
For single triangle:
Circuit
≅ Overall one Trotter step looks like:
3 Layers of operations Easy to extend to larger 2d lattice space
E.g. 2x2 torus
Just duplicate E and XY circuits.
The plaquettes with the same color commute on their own.
→ Only one more layer of plaquette operations. On the IBM Q “Johannesburg” device
i
: i-th physical qubit
: coulings between qubits On the IBM Q “Johannesburg” device
0
3
1
2
5
4 On the IBM Q “Johannesburg” device
0
3
1
2
5
4 On the IBM Q “Johannesburg” device
0
3
1
2
5
4 On the IBM Q “Johannesburg” device
0
3
1
2
5
4 On the IBM Q “Johannesburg” device
0
1
3
2
5
4 On the IBM Q “Johannesburg” device
PD of measuring the
state
|ψ>=e^(-iHBt)x
e^(-iHXYt)e^(-iHEt)|0
>
with t=0.1
IBM Q
Exact Error mitigation
noisy circuit
mitigation
operation
Error ε is mitigated with its inverse operation ε-1 Error mitigation
Mitigated Trotter: Mathematical Formalism of Error mitigation
Error modeling:
Measurement of
noisy circuit
Characterization
of the noise
Measurement of
the state
UρiU†with some
basis |k> and ref
state ρi Mathematical Formalism of Error mitigation
Error modeling:
From experiments
Want
Computable
Measurement of
noisy circuit
Characterization
of the noise
Measurement of
the state
UρiU†with some
basis |k> and ref
state ρi Mathematical Formalism of Error mitigation
Error modeling:
From experiments
Want
Computable
Measurement of
noisy circuit
Characterization
of the noise
Measurement of
the state
UρiU†with some
basis |k> and ref
state ρi Mathematical Formalism of Error mitigation
Compute the mitigation operation:
-1
Since ε ○ ε is identity,
Want
Known Mathematical Formalism of Error mitigation
-1
Now we know ε . It can be expressed as Mathematical Formalism of Error mitigation
-1
Now we know ε . It can be expressed as
We approximate this by taking only the
diagonal elements for simplicity: Mathematical Formalism of Error mitigation
Ideal probability measuring x can be
obtained from the LC of the noisy circuits
with some additional Pauli: Error mitigation
In our case, we have multiple mitigation operations,
but the idea is the same. On the IBM Q “Johannesburg” device
PD of measuring the
state
|ψ>=e^(-iHBt)x
e^(-iHXYt)e^(-iHEt)|0
>
with t=0.1
With mitigation
No mitigation On the IBM Q “Johannesburg” device
PD of measuring the
state
|ψ>=e^(-iHBt)x
e^(-iHXYt)e^(-iHEt)|0
>
with t=0.1
With mitigation
Exact What I have to work on from now
● Pauli Twirling to restrict ε-1 to have exactly only
diagonal elements
d+1-dim Gravity
d-dim GT What I have to work on from now
● Pauli Twirling to restrict ε-1 to have exactly only
diagonal elements
● Does there exist a duality to a strongly coupled
spin system? Analogous to Ads/CFT duality:
d+1-dim Gravity
Weak Coupling LGT
d-dim GT
Strong Coupling
Spin System Summary
● Investigated the lattice gauge theory (2+1d U(1)
theory) model on a spin basis.
● Constructed an optimized quantum circuit for it.
● Tested it on a noisy quantum device with the noise
mitigation which gave improvements in the
simulation result.
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