Spectral Gap Scaling of One Dimensional Quantum Spin Chains

Spectral Gap Scaling of One Dimensional Quantum Spin Chains

A talk from the 2012 Caltech Summer Undergraduate Research Fellowship Perpall Speaking competition. I ended up placing in the top 7.

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Travis Scholten

January 19, 2012
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  1. 1.

    Spectral Gap Scaling - 1D Spin Chains 1/13 Spectral Gap

    Scaling of One Dimensional Quantum Spin Chains Travis Scholten, Caltech Robert L. Blinkenberg SURF Fellow January 19, 2012 Mentors: John Preskill & Spyridon Michalakis Additional Help: Steve Flammia (UWashington) Matthew Heydeman (Caltech UG) Frank Rice (Caltech Physics)
  2. 2.

    Spectral Gap Scaling - 1D Spin Chains 2/13 Outline 1

    Introduction - Where did we come from? 2 Problem Statement - What should we solve? 3 Results - What did we solve? 4 Summary/Questions - Where are we going?
  3. 3.

    Spectral Gap Scaling - 1D Spin Chains 3/13 Introduction -

    Where did we come from? Computer Science & Satisfiability Problems – Satisfiability problem: S = C1 ∧ C2 · · · ∧ CM – Each clause Ci is a Boolean function of bits: Ci = z1 ∨ ¬z4 ∨ z2 · · · – Goal: Pick bit values so S evaluates to “true” example: S = z1 ∧ z2 =⇒ z1, z2 = ? – Famous problems: 2-sat & 3-sat
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    Spectral Gap Scaling - 1D Spin Chains 4/13 Introduction -

    Where did we come from? Physics & Qubits – Turn S into another kind of problem: - Create problem Hamiltonian HP with quantum bits HP = i hCi hCi = 1 if clause Ci is violated 0 if clause Ci is satisfied - Introduce quantum state |Q = |z1z2z3 · · · zM – Satisfying S ⇐⇒ find qubits so HP |Q = 0 – How do we find |Q ?
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    Spectral Gap Scaling - 1D Spin Chains 5/13 Introduction -

    Where did we come from? Adiabatic Quantum Computation (AQC) - Start: initial Hamiltonian HI - ground state easy to prepare - Finish: problem Hamiltonian HP - ground state encodes satisfying assignment Excited State Energy Ground State Energy 0.2 0.4 0.6 0.8 1.0 s 0.1 0.2 0.3 0.4 0.5 0.6 Energy – Interpolate: H(s) = (1−s)HI +sHP (s = t/T) – Searching for a ground state... – What’s with T?
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    Spectral Gap Scaling - 1D Spin Chains 6/13 Introduction -

    Where did we come from? A Trick With Time – Adiabatic Theorem: T → ∞ =⇒ state is |Q – If T >> E g2 min then state “close to” |Q – E scaling: – gmin scaling: ? – What’s our model to be?
  7. 7.

    Spectral Gap Scaling - 1D Spin Chains 7/13 Problem Statement

    - What should we solve? Quantum 2-sat – Consider a generic Hamiltonian on a ring H = N k=1 |Ψ Ψ|k,k+1 =⇒ H 1 2 N – How does gmin scale with system size N for arbitrary |Ψ ? – What about s?
  8. 8.

    Spectral Gap Scaling - 1D Spin Chains 8/13 Results -

    What did we solve? Step 1: Fewer Numbers – General quantum state has 8 real parameters! – Transform state to give spectrally equivalent Hamiltonian – 3 real parameters: 0 ≤ λ ≤ 1 0 ≤ α ≤ 1 − 2π ≤ δ < 2π – Any path depending on s lies in here. – What is gmin?
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    Spectral Gap Scaling - 1D Spin Chains 9/13 Results -

    What did we solve? Step 2: Use a Computer – Numerics Algorithm: - Choose λ, α, δ - For 10 ≤ N ≤ Nmax compute gmin for N qubits - Curve fitting: gmin versus N - Repeat – Are we on the right track? λ α Phase Diagram for δ=1 (in units of π) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 <−−−−−−Faster Slower−−−−−−> 1 1.5 2 2.5 3 3.5 4
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    Spectral Gap Scaling - 1D Spin Chains 10/13 Results -

    What did we solve? Checking Our Work – Pick λ = 1 √ 2 and α = δ = 0 to obtain: H = − 1 4 N k=1 (X ⊗ X + Y ⊗ Y + Z ⊗ Z)k,k+1 =⇒ spin-1/2 Heisenberg ferromagnet on a ring – Hans Bethe (1931): Analytical solution, predicts gmin(N) = 1 4 1 − cos 2π N – Do numerics agree?
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    Spectral Gap Scaling - 1D Spin Chains 11/13 Results -

    What did we solve? Our Result - Agreement! 0 0.05 0.1 0.15 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 1−cos(2π/N) g min Spectral Gap Scaling − Heisenberg Ferromagnet g min (N) = 0.245[1−cos(2π/N)] + 0.000114
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    Spectral Gap Scaling - 1D Spin Chains 12/13 Summary/Questions -

    Where are we going? Summary Known Knowns – From Computer Science to Physics – 2-qubit projection operators & spin models – Numerics code: Known Unknowns – Is AQC poly-time for all such projection models? – Extend results to more general cases? (non-nearest neighbor) Unknown Unknowns – Questions?
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    Spectral Gap Scaling - 1D Spin Chains 13/13 Summary/Questions -

    Where are we going? References - Helpful Reading –Bravyi, Sergey: “Efficient algorithm for a quantum analogue of 2-SAT” arXiv.org:quant-ph/0602108 –Farhi, Edward et. al: “Quantum Computation by Adiabatic Evolution” arXiv.org:quant-ph/0001106v1 –Karbach, Michael et. al: “Introduction to the Bethe Ansatz I” arXiv.org:cond-mat/9809162v1