qft simulation in the age of quantum computersphysics.sharif.edu/~qc/files/slides-3.pdfgross, d. j....
TRANSCRIPT
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QFT Simulation in the Age of Quantum Computers
By: Ali Hamed Moosavian
Date: July 23rd, 2020
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Quantum Field Theory
● Much is known about using quantum computers to simulate quantum systems.
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Quantum Field Theory
● Much is known about using quantum computers to simulate quantum systems.
● Why might Quantum Field Theory be different?
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Quantum Field Theory
● Much is known about using quantum computers to simulate quantum systems.
● Why might Quantum Field Theory be different?– Field has infinitely many degrees of freedom– Relativistic– Particle number not conserved– Formalism looks different
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Representing Quantum Fields
● A field is a list of values, one for each location in space.
● A quantum field is a superposition over classical fields.
● A superposition over bit strings is a state of a quantum computer.
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Classical Algorithms
● Feynman Diagrams
● Breaks down at strong coupling or high precision
● Lattice Methods
● Good for binding energies.
● Real-time dynamics difficult
There’s room for exponential speedup by quantum computing.
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Layout of the scattering Problem
● Input:– A set of incoming particles
with given momenta
– A Hamiltonian
– A time of evolution
● The Output of Scattering:– A set of particles with their
momenta at the end of the evolution time
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Bosonic vs Fermionic
● Boson statistics:– Map to local operators– They can all occupy
the same quantum state.
– The number of qubits per lattice site is proportional to the energy cutoff.
● Fermion statistics– Map to non-local
operators● Jordan-Wigner
transformation● Workarounds exist
– Can be represented with a constant number of qubits per site.
– Fermion doubling
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Motivation
● Jordan-Lee-Preskill (2014)– Proof of concept for
BQP– Run-time dominated by
state preparation– State preparation very
slow– Limited to the phase
directly connected to free theory
● Two paths forward to improve state preparation– More efficient state
preparation algorithms– More general state
preparation algorithms
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Gross-Neveu Model
● Toy model for QCD● 1+1d- fermionic● Asymptotic
Freedom● Has flavors● Chiral symmetry
– Broken by mass term
Gross, D. J. & Neveu, A. Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D 10, 3235–3253 (1974).
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1.Prepare the GS of system, NN-term and g0 set to zero
2.Adiabatically turn on the NN-term
3.Adiabatically turn on the g0 term
4.Excite particles by adding a weak sinusoidal source term to the Hamiltonian.
5.Evolve in time
6.Either phase estimate local charges or adiabatically return to free theory then phase estimate number operators of momentum mode.
Jordan-Lee-Preskill(2014)
Jordan, S. P., Lee, K. S. M. & Preskill, J. Quantum Algorithms for Fermionic Quantum Field Theories. 29 (2014).
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Fermion Doubling Problem
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Wilson term
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Faster quantum algorithm to simulate fermionic quantum field theory
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Faster quantum algorithm to simulate fermionic quantum field theory
● Discretize and put on a lattice.● Use DMRG to calculate the GS of interacting theory.
– Map the MPS to a quantum circuit.
– Apply the Quantum circuit to get the GS.● Excite particles using Rabi Oscillations.
– Floquet’s Theorem, guarantees excitation.● Evolve in time(e.g. Suzuki-Trotter)● Measure the outcome using phase estimation algorithm.● Performance, Quantum: Classical:
Hamed Moosavian, A. & Jordan, S. Faster quantum algorithm to simulate fermionic quantum field theory. Phys. Rev. A 98, 012332 (2018).
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From MPS to Quantum Circuit
● DMRG:
● SVD:
Schön, Hammerer, Wolf, Cirac & Solano Phys. Rev. A 75, 032311 (2007).
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From MPS to Quantum Circuit
● Quantum Circuit:
● Each of these classical steps, run in time:
Schön, Hammerer, Wolf, Cirac & Solano Phys. Rev. A 75, 032311 (2007).
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Exciting Initial Particles● Starting in the ground state of interacting
Hamiltonian, simulate dynamics with a sinusoidal source term(Rabi Oscillations):
● is chosen so it resonates with our desired state:
● Ensure the desired momentum with W:
Hamed Moosavian, A. & Jordan, S. Faster quantum algorithm to simulate fermionic quantum field theory. Phys. Rev. A 98, 012332 (2018).
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Few-Level Approximation
Hamed Moosavian, A. & Jordan, S. Faster quantum algorithm to simulate fermionic quantum field theory. Phys. Rev. A 98, 012332 (2018).
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Two-Level Approximation
Hamed Moosavian, A. & Jordan, S. Faster quantum algorithm to simulate fermionic quantum field theory. Phys. Rev. A 98, 012332 (2018).
● ↓ν and ↑δ = better approximation
● ν is the degeneracy in the rotating frame = number of states on resonance
● 2-levels for anharmonic asymmetric case.
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Applying Floquet’s Theorem
Hamed Moosavian, A. & Jordan, S. Faster quantum algorithm to simulate fermionic quantum field theory. Phys. Rev. A 98, 012332 (2018).
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Exciting Initial Particles, Bounds● Two ingredients go into our analysis of the
success rate of exciting particles:
1. 2-level approximation. Error bound:
2. Analyze 2-level system with Floquet’s theory:
Moosavian, A. H. & Jordan, S. http://arxiv.org/abs/1711.04006 (2017).
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Limitations
● Jordan-Lee-Preskill:– Not practical:
– Adiabatic state preparation:
● Limited to the phase adiabatically connected to the free phase of the theory.
● Moosavian-Jordan:– 1+1 space-time
dimensions– Bottlenecked by
classical heuristic algorithm:
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Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories
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Overview of the Algorithm
● Start with the ground state of a small system:– Can be explicitly calculated
● We add sites to the ground state, one site at a time● Two requirements:
– The spectral gap doesn’t close:● Mass gap
– The inner product between consecutive ground states is not exponentially close to zero:
● Physical intuition: with finite correlation length, the inner product should approach a constant.
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Jagged Adiabatic Path Lemma
Aharonov, D. & Ta‐Shma, A. Adiabatic Quantum State Generation. SIAM J. Comput. 37, 47–82 (2007).
H1
H2
H3
HT
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Preparing PEPS on a QC
Schwarz, M., Temme, K. & Verstraete, F. Preparing Projected Entangled Pair States on a Quantum Computer. Phys. Rev. Lett. 108, 110502 (2012).
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Modified Jagged Path Lemma
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
● Sequence of geometrically local, bounded norm, gapped Hamiltonians
● Non-vanishing overlap of ground states
● Predictions of GS energies
➔ QA in run-time
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The Algorithm
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
● Set Lattice Spacing ● Discretize the Hamiltonian● Prepare the GS of a small system● Apply a unitary, e.g. Hadamard on the rest of
the system● Repeat:
– Transform to using Theorem 1
● Runtime:
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Adding sites
● Keep the shape as close to a D-dimensional hypercube as possible.
● This ensures the overlaps do not vary too much during the preparation algorithm.
● If correlation length is finite, only a finite amount of sites will be affected.
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The Conjecture
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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The Conjecture
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
● Proven cases:– GS is injective PEPS
Phys. Rev. Lett. 108, 110502 (2012).
– GS is topological PEPS
Phys. Rev. A - At. Mol. Opt. Phys. 88, 1–5 (2013).
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The Conjecture
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
● Proven cases:– GS is injective PEPS
Phys. Rev. Lett. 108, 110502 (2012).
– GS is topological PEPS
Phys. Rev. A - At. Mol. Opt. Phys. 88, 1–5 (2013).
● Counter example:– AKLT
● Does not have a single site coarse continuum limit!
● Solution:– Add two sites at a
time
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Testing on Classical Computers
● Gross-Neveu model● Pick some reasonable variables:
–
● Calculate GS using DMRG for different system sizes:– up to 50 sites
● Calculate inner product between GS of consecutive systems.
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Mass Renormalization & Correlation Length● How to calculate the gap? How about CL?● Continuous Free Theory:
● Discretized and Interacting Theory:
– Where m is the renormalized mass and is inversely proportional to the correlation length χ.
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Choosing an appropriate parameters
● A reasonable correlation length should be much larger than lattice spacing and much smaller than system size at the same time.
● Preliminary numerical investigation:
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Two point correlation functions
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Correlation lengths
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Inner Product
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Predicting Ground Energies
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
● Idea: Use previous energy data to predict the next one.
● Two models were tried:– Constant energy density– Energy density with Casimir effect corrections
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Predicting Ground Energies
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Predicting Ground Energies
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Summary & Conclusion
● We’ve looked at two state preparation algorithms– MPS– Site-by-site
● Direct applications in simulation of fermionic QFTs.● Other possible applications:
– Optimization problems– Quantum chemistry algorithms
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Open Problems
● Future ideas:– Prove the conjecture– Check with other QFT– Check if they can run on NISQ era quantum
computers– Improve upon bosonic algorithms– Quantum Algorithm for hybrid systems– The Standard Model
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Thank you
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Bond Dimension
● It suffices to take where errors shrink superpolynomially with k, resulting in
● For lattice spacing small compared to the correlation length, can be estimated from a CFT argument.
● , therefore, the complexity for preparing the interacting vacuum is:
Swingle, B. https://arxiv.org/abs/1304.6402 (2013).
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Efficient Phase Estimation
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).
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Mapping Overlapping GS
Hamed Moosavian, A., Garrison, J. R. & Jordan, S. P. Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories. (2019).39. Yoder, T. J., Low, G. H. & Chuang, I. L. Fixed-Point Quantum Search with an Optimal Number of Queries. Phys. Rev. Lett. 113, 210501 (2014).
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DMRG package in Julia