3 June 2020 Semina for RCCS Lattice gauge theory with quantum computers Akio Tomiya (RIKEN-BNL) akio.tomiya@riken.jp T. Izubuchi, Y. Kikuchi (RBRC) M. Honda (YITP) B. Chakraborty (Cambridge) arXiv: 2001.00485
Who am I? Akio Tomiya Riken/BNL, particle physics, Lattice QCD, and ML Publications Biography 2018 - : Postdoc in RIKEN-BNL (NY, US) 2015 - 2018 : Postdoc in CCNU (Wuhan, China) 2015 : PhD at Osaka RCCS seminar Schwinger model with QC 2
Outline 1. The sign problem in Quantum field theory 4P 2. Quantum computer 7P 3. Schwinger model with lattice-Hamiltonian formalism 10P 4. Adiabatic preparation of vacuum 3P 5. Results 6P 3
Outline 1. The sign problem in Quantum field theory 4P 2. Quantum computer 7P 3. Schwinger model with lattice-Hamiltonian formalism 10P 4. Adiabatic preparation of vacuum 3P 5. Results 6P 4
Motivation, Big goal Akio Tomiya Non-perturbative calculation of QCD is important QCD in 3 + 1 dimension S = ∫ d 4 x [ − 1 / − m ) ψ ] 4 tr F μν F μν + ¯ ψ ( i ∂ / − gA Z = ∫ A ¯ ψ ψ e i S F μν = ∂ μ A ν − ∂ ν A μ − i g [ A μ , A ν ] • This describes… d D- 0 • inside of hadrons (bound state of quarks), mass of them K c s • scattering of gluons, quarks - e W • Equation of state of neutron stars, Heavy ion collisions, − ν etc • Non-perturbative e ff ects are essential . How can we deal with? • Confinement • Chiral symmetry breaking RCCS seminar Schwinger model with QC 5
Motivation, Big goal Akio Tomiya LQCD = Non-perturbative calculation of QCD QCD in 3 + 1 dimension S = ∫ d 4 x [ − 1 / − m ) ψ ] 4 tr F μν F μν + ¯ ψ ( i ∂ / − gA Z = ∫ A ¯ ψ ψ e i S F μν = ∂ μ A ν − ∂ ν A μ − i g [ A μ , A ν ] QCD in Euclidean 4 dimension S = ∫ d 4 x [ + 1 / − m ) ψ ] 4 tr F μν F μν + ¯ ψ ( ∂ / − gA Z = ∫ A ¯ ← This can be regarded ψ ψ e − S as a statistical system • Standard approach: Lattice QCD with Imaginary time and Monte-Carlo • LQCD = QCD + cuto ff + irrelevant ops. = “Statistical mechanics” • Mathematically well-defined quantum field theory • Quantitative results are available = Systematic errors are controlled RCCS seminar Schwinger model with QC 6
Motivation, Big goal Akio Tomiya Sign problem prevents using Monte-Carlo • Monte-Carlo is very powerful method to evaluate expectation values for “statistical system”, like lattice QCD in imaginary time N conf 1 1 U c ← P ( U ) = 1 ∑ Z e − S [ U ] ⟨ O [ U ] ⟩ = O [ U c ] + 𝒫 ( ) ∈ ℝ + N conf N conf c Great successes! Sign problem arXiv:0906.3599 • However, if we have, real time, finite theta, finite baryon density case, we cannot we use Monte-Carlo technique because e^{-S} becomes complex. This is no more probability. • Hamiltonian formalism does not have such problem! But it requires huge memory to store quantum states, which cannot realized even on supercomputer. • Quantum states should not be realized on classical computer but on quantum computer (Feynman 1982) RCCS seminar Schwinger model with QC 7
Short summary Akio Tomiya Sign problem prevent to use conventional method • QCD describes perturbative and non-perturbative phenomena • Lattice QCD with imaginary time is non-perturbative and quantitive method, which is evaluated by Monte-Carlo • Sign problem, which is occurred in real time/finite theta/finite baryon density cace, prevents us to use the Monte-Carlo • Hamiltonian formalism is one solution but we cannot construct the Hilbert space because of the dimensionality • Quantum simulation/computer is natural realization the Hamiltonian formalism Question? RCCS seminar Schwinger model with QC 8
Outline 1. The sign problem in Quantum field theory 4P 2. Quantum computer 7P 3. Schwinger model with lattice-Hamiltonian formalism 10P 4. Adiabatic preparation of vacuum 3P 5. Results 6P 9
Quantum computer? Akio Tomiya Towards beyond classical computers Quantum 1946 IBM Q Classical | 0 ⟩ , | 1 ⟩ 0,1 State Data → Machine → State → Machine → Data Quantum Classical computer Unit Qubit Bit Unitary Operation Logic gates | ↑ ⟩ , | ↓ ⟩ operations Represent Spin Voltage ation of 0/1 High, low Glowing Neven’s law Moor’s law law double exp(?) exp Lattice gauge theory with quantum computer could be a future “common tool” https://uk.pcmag.com/forward-thinking/117979/gartners-top-10-strategic-technology-trends RCCS seminar Schwinger model with QC 10
⃗ Quantum computer? Akio Tomiya We can perform bit operation + α Classical “NOT” 011 100 Quantum “NOT” | 011 ⟩ = | 0 ⟩ ⊗ | 1 ⟩ ⊗ | 1 ⟩ | 100 ⟩ = | 1 ⟩ ⊗ | 0 ⟩ ⊗ | 0 ⟩ In addition, 1 “Entangling” CN 10 H 0 | 0 ⟩ ⊗ | 0 ⟩ = | 0 ⟩ ⊗ ( | 0 ⟩ + | 1 ⟩ ) | 0 ⟩ ⊗ | 0 ⟩ 2 1 ( | 00 ⟩ + | 11 ⟩ ) = 2 But, what is benefit for physicist? RCCS seminar Schwinger model with QC 11
Quantum computer? Akio Tomiya For physicists : Circuit ~ time evolution of quantum spins Example1. Transverse Ising model on 3 sites (Open boundary) : Pauli matrix of x on site j X j H = − ∑ Z j Z k − h ∑ X j = − Z 0 Z 1 − Z 1 Z 2 − hX 0 − hX 1 − hX 2 : Pauli matrix of z on site j Z j < j , k > j : size of external field h Time evolution for infinitesimal (real) time ε : e − i H ϵ = e − i( − Z 0 Z 1 − Z 1 Z 2 − hX 0 − hX 1 − hX 2 ) ϵ ≈ e − i( − Z 0 Z 1 − Z 1 Z 2 ) ϵ e − i( − hX 0 − hX 1 − hX 2 ) ϵ + O ( ϵ 2 ) (Suzuki-Trotter expansion) ≡ U ZZ ( ϵ ) ≡ U X ( ϵ ) Qubit = spin Unitary trasformation on a qubit = gate | 0 ⟩ = | ↑ ⟩ R Z ( θ ) | ψ ⟩ = e − i 1 2 θ Z | ψ ⟩ ~ Hamiltonian evol. | 1 ⟩ = | ↓ ⟩ We can make these boxes from gates (ask me later) | ↑ ⟩ e − i Ht | ↑ ⟩ ⊗ | ↑ ⟩ ⊗ | ↑ ⟩ = ⋯ = | Ω ( t ) ⟩ U ZZ ( ϵ ) U X ( ϵ ) U ZZ ( ϵ ) U X ( ϵ ) | ↑ ⟩ = | Ω (0) ⟩ | ↑ ⟩ In this way, we can (re)produce, Hamiltonian time evolution using a quantum circuit. Here we can evaluate the systematic error from the expansion and reduce it by using higher order decomposition (leapfrog etc) Quantum computer actually can realize any unitary transformation (skipping proof) RCCS seminar Schwinger model with QC 12
Quantum computer? Akio Tomiya For physicists : Circuit ~ time evolution of quantum spins Example2 We can make wave functional for a given Hamiltonian for 2nd quantized system z counter | ↑ ⟩ … U ZZ ( ϵ ) U X ( ϵ ) U ZZ ( ϵ ) U X ( ϵ ) z counter | ↑ ⟩ Collapse of state | Ω ( t ) ⟩ regard as z counter measures spins up/down in probability (Born rule), many trial gives histogram: # of ↑ ↑ ∝ | ⟨ ↑ ↑ | Ω ( t ) ⟩ | 2 count # of ↑ ↓ ∝ | ⟨ ↑ ↓ | Ω ( t ) ⟩ | 2 # of ↓ ↑ ∝ | ⟨ ↓ ↑ | Ω ( t ) ⟩ | 2 # of ↓ ↓ ∝ | ⟨ ↓ ↓ | Ω ( t ) ⟩ | 2 ↑ ↑ ↑ ↓ ↓ ↑ ↓ ↓ On the other hand, the magnetization is, 2 2 ∑ ∑ ∑ (insert complete set) ⟨Ω ( t ) | Z k | Ω ( t ) ⟩ = ⟨Ω ( t ) | Z k | Ψ⟩⟨Ψ | Ω ( t ) ⟩ k =1 k =1 Ψ = ↑↑ , ⋯ Ψ = ↑ ↑ , ↑ ↓ , ↓ ↑ , ↓ ↓ 2 ∑ ∑ ( − 1) Ψ k | ⟨Ψ | Ω ( t ) ⟩ | 2 (can be constructed from data) = Ψ k = spin on site k k =1 Ψ = ↑↑ , ⋯ We can calculate expectation values! RCCS seminar Schwinger model with QC 13
Quantum computer? Akio Tomiya Quantum computer is under developing Quantum computer is theoretically universal, namely it can mimic any unitary transformation, but practically … 1. The number of qubits are not many. up to 53 (world record) U ZZ ( ϵ ) U X ( ϵ ) 2. Gate operations are inaccurate. = We cannot make quantum circuit deeper. e.g.) Control-not (CNOT) gate If ● side is 0, gate does nothing on the target ⊕ If ● side is 1, gate flips the target ⊕ side. | Ψ⟩ | Ψ⟩ controller { | 0 ⟩ (if Ψ = 0) | 1 ⟩ (if Ψ = 1) | 0 ⊕ Ψ⟩ = | 0 ⟩ target (machine dependent, 1903.10963) | actual ⟨ 0 ⊕ Ψ | 0 ⊕ Ψ⟩ ideal | ∼ 0.97 Operations are sometimes wrongly performed. In order to study machine independent parts, we use a simulator instead of real one. RCCS seminar Schwinger model with QC 14
Quantum computer? Akio Tomiya IBM Q is available and free From Jupyter/python From Browser to real machine Several frameworks are available; Qiskit : de facto standard (IBM) Qulacs : Fastest simulator (QunaSys, Japan) Blueqat : I think this is easiest (MDR, Japan) etc… RCCS seminar Schwinger model with QC 15
Short summary Akio Tomiya Quantum computer? • Quantum computer is developing technology. Current one is noisy so far • Once hamiltonian is constructed, we can perform time evolution using quantum circuit in principle • Comment1: We use simulator but our technology can be used in future machines with error-correction. Time resolves this problem. • Comment2: Simulation of quantum computer by classical machine is generally exponentially hard. To calculate large problem, we need real device. Question? RCCS seminar Schwinger model with QC 16
Outline 1. The sign problem in Quantum field theory 4P 2. Quantum computer 7P 3. Schwinger model with lattice-Hamiltonian formalism 10P 4. Adiabatic preparation of vacuum 3P 5. Results 6P 17
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