On complexity of the quantum Ising model Sergey Bravyi (IBM) - PowerPoint PPT Presentation

On complexity of the quantum Ising model Sergey Bravyi (IBM) Matthew Hastings (Microsoft Research) Presenter: David Gosset Based on QIP arXiv: 1410.0703 Sydney arXiv: 1402.2295 January 16, 2015

Motivation Quantum annealing with >100 qubits Quantum Hamiltonian Complexity Boixo et al, Nature Phys. 10, 218 (2014) Cubitt & Montanaro, arxiv:1311.3161 QMA TIM Attempts to solve hard optimization NP problems such as QUBO P Basic model of phase transitions Onsager (1944) Understand computational hardness of estimating the ground state energy for quantum spin Hamiltonians

Transverse Ising Model ( TIM ) Qubits live at vertices of a graph • Ising 𝑎𝑎 interactions between • nearest neighbor qubits. Local magnetic fields along • 𝑌 and 𝑎 axes. 𝐼 = 𝑕 𝑣 𝑎 𝑣 + ℎ 𝑣 𝑌 𝑣 − 𝐾 𝑣,𝑤 𝑎 𝑣 𝑎 𝑤 𝑣 (𝑣,𝑤) 𝑎 𝑣 = 1 0 𝑌 𝑣 = 0 1 0 −1 1 0

Part I Universality of TIM for quantum annealing Part II Computational hardness of estimating the ground state energy of TIM Part III Ferromagnetic TIM is easy

Quantum Annealing (Farhi et al 2001) 𝑗 𝜖| Ψ(𝑢) = 𝐼(𝑢/𝑈)| Ψ(𝑢) 𝜖𝑢 Hard Easy Unitary evolution 0 ≤ 𝑢 ≤ 𝑈 Easy: 𝐼 0 = − 𝑣 𝑌 𝑣 Hard: 𝐼 1 = (𝑣,𝑤) 𝐾 𝑣,𝑤 𝑎 𝑣 𝑎 𝑤 + 𝑣 𝑕 𝑣 𝑎 𝑣 𝐼(𝑡) interpolates between 𝐼 0 and 𝐼(1)

Given an adiabatic path 𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1, how large the evolution time 𝑈 should be ? Adiabatic Theorem 𝐼 ∥ 2 𝑈~ ∥ 𝐼 ∥ + ∥ + ∥ 𝐼 ∥ 𝜀 2 𝜀 3 𝜀 2 Here 𝜀 is the minimum spectral gap above the ground state of 𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1. Jansen, Seiler, Ruskai, JMP 48, 102111 (2007) We need a smooth path with a non-negligible spectral gap

Big open question: what kind of problems can be efficiently solved by the quantum annealing (QA) ?

Big open question: what kind of problems can be efficiently solved by the quantum annealing (QA) ? Simpler question: can one QA machine efficiently simulate another QA machine ? target QA machine simulator QA machine 𝑁′ 𝑁

Big open question: what kind of problems can be efficiently solved by the quantum annealing (QA) ? Simpler question: can one QA machine efficiently simulate another QA machine ? target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians

Big open question: what kind of problems can be efficiently solved by the quantum annealing (QA) ? Simpler question: can one QA machine efficiently simulate another QA machine ? target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians Some fixed target class of Hamiltonians

What does efficient simulation mean ? Target Simulator 𝐼 ′ 𝑡 , 0 ≤ 𝑡 ≤ 1 𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1 Adiabatic path 𝑜 ′ ≤ 𝑞𝑝𝑚𝑧(𝑜) 𝑜 Number of qubits 𝜀 ′ ≥ 𝜀 𝜀 Minimum spectral gap 𝐾 ′ ≤ 𝑞𝑝𝑚𝑧(𝑜, 𝜀 −1 , 𝐾) Maximum 𝐾 interaction strength All spins | All spins | + + Ground state at 𝑡 = 0 Ground state at 𝑡 = 1 | ≈ 𝑊| 𝜔 𝜔 Here 𝑊: 𝐃 2 ⨂𝑜 → 𝐃 2 ⨂𝑜′ is a sufficiently simple encoding

When efficient simulation is unlikely: target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians 2-local Hamiltonians

When efficient simulation is unlikely: target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians 2-local Hamiltonians BQP Aharonov et al (2007) Oliveira and Terhal (2008)

When efficient simulation is unlikely: target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians 2-local Hamiltonians BQP BQP⋂postBPP SB, DiVincenzo, Oliveira, Aharonov et al (2007) Terhal (2007) Oliveira and Terhal (2008)

When efficient simulation is unlikely: target QA machine simulator QA machine 𝑁′ 𝑁 TIM Hamiltonians 2-local Hamiltonians More Powerful ≠ BQP BQP⋂postBPP SB, DiVincenzo, Oliveira, Aharonov et al (2007) Terhal (2007) Oliveira and Terhal (2008)

Stoquastic k-local Hamiltonians System of 𝑜 qubits with a Hamiltonian 𝐼 = 𝐼 𝛽 𝛽 Each term 𝐼 𝛽 acts on at most 𝑙 = 𝑃(1) qubits 1. Matrix elements of 𝐼 𝛽 in the standard basis are real. 2. Off-diagonal matrix elements of 𝐼 𝛽 are non-positive: 𝑦|𝐼 𝛽 |𝑧 ≤ 0 for all 𝑦 ≠ 𝑧 ∈ 0,1 𝑙

Building blocks for 2-local stoquastic Hamiltonians: Diagonal : ±𝑎 𝑣 , ±𝑎 𝑣 𝑎 𝑤 Transverse field: −𝑌 𝑣 0 0 , Elementary −𝑌 ⊗ −𝑌 ⊗ 1 1 interactions: − 𝑌 ⊗ 𝑌 − 𝑍 ⊗ 𝑍, −𝑌 ⊗ 𝑌 + 𝑍 ⊗ 𝑍

Result 1: universality of TIM for quantum annealing with 2-local stoquastic Hamiltonians simulator QA machine target QA machine = 𝑁′ 𝑁 TIM Hamiltonians Stoquastic 2-local Hamiltonians

Result 1: universality of TIM for quantum annealing with 2-local stoquastic Hamiltonians simulator QA machine target QA machine = 𝑁′ 𝑁 TIM Hamiltonians Stoquastic 2-local Hamiltonians with k-local diagonal terms

Result 1: universality of TIM for quantum annealing with 2-local stoquastic Hamiltonians simulator QA machine target QA machine = 𝑁′ 𝑁 TIM Hamiltonians Stoquastic 2-local Hamiltonians on degree-3 graphs with k-local diagonal terms

Part II Computational hardness of estimating the ground state energy of TIM

Ground state energy: 𝐹 0 = min 𝜔 𝐼 𝜔 Local Hamiltonian Problem (LHP): Input: (𝑜, 𝐼 = 𝛽 𝐼 𝛽 , 𝐷 𝑧𝑓𝑡 < 𝐷 𝑜𝑝 ) Yes-instance: 𝐹 0 ≤ 𝐷 𝑧𝑓𝑡 No-instance: 𝐹 0 ≥ 𝐷 𝑜𝑝 Decide which one is the case. Promise: 𝐹 0 ∉ 𝐷 𝑧𝑓𝑡 , 𝐷 𝑜𝑝 Normalization: 𝐼 𝛽 ≤ 𝑞𝑝𝑚𝑧 𝑜 , 𝐷 𝑜𝑝 − 𝐷 𝑧𝑓𝑡 ≥ 𝑞𝑝𝑚𝑧 1/𝑜 #terms ≤ 𝑞𝑝𝑚𝑧(𝑜)

Merlin-Arthur games (Babai 1985) I instance of yes/no problem P Merlin Arthur proof Polynomial-time Unlimited classical computer computational power reject accept

A problem belongs to this class if … complexity class yes-instance : Arthur accepts some Merlin’s proof NP no-instance: Arthur rejects any Merlin’s proof

A problem belongs to this class if … complexity class yes-instance : Arthur accepts some Merlin’s proof NP no-instance: Arthur rejects any Merlin’s proof Arthur is a quantum computer. Merlin’s proof can be a quantum state. QMA yes-instance: Arthur accepts some Merlin’s proof with high probability no-instance: Arthur rejects any Merlin’s proof with high probability

A problem belongs to this class if … complexity class yes-instance : Arthur accepts some Merlin’s proof NP no-instance: Arthur rejects any Merlin’s proof Arthur is a quantum computer. Merlin’s proof can be a quantum state. QMA yes-instance: Arthur accepts some Merlin’s proof with high probability no-instance: Arthur rejects any Merlin’s proof with high probability Same as QMA but Arthur can apply only reversible StoqMA classical gates (CNOT, TOFFOLI) and measure some fixed output qubit in the X-basis. Arthur accepts the proof if the measurement outcome is ‘ +′ . Arthur can use | 0 and | + ancillas. SB, Bessen, Terhal, arXiv:0611021

Π 2 PostBPP A 0 PP AM QMA SBP StoqMA MA - randomized analogue of MA NP NP AM=MA + shared randomness SBP approximate counting classes P A 0 PP

Computing the minimum energy of the classical Ising model is NP -complete, even for the 2D geometry (with magnetic field) Barahona (1982)

Computing the minimum energy of the classical Ising model is NP -complete, even for the 2D geometry (with magnetic field) Barahona (1982) Local Hamiltonian Problem for general 𝑙 -local Hamiltonians is QMA -complete for any constant 𝑙 ≥ 2 Kitaev, Kempe, Regev (2006); QMA -complete for the 2D geometry Oliveira and Terhal (2008)

Computing the minimum energy of the classical Ising model is NP -complete, even for the 2D geometry (with magnetic field) Barahona (1982) Local Hamiltonian Problem for general 𝑙 -local Hamiltonians is QMA -complete for any constant 𝑙 ≥ 2 Kitaev, Kempe, Regev (2006); QMA -complete for the 2D geometry Oliveira and Terhal (2008) Local Hamiltonian Problem for 𝑙 -local stoquastic Hamiltonians is StoqMA -complete for any constant 𝑙 ≥ 2 SB, DiVincenzo, Oliveira, Terhal (2007)

Computing the minimum energy of the classical Ising model is NP -complete, even for the 2D geometry (with magnetic field) Barahona (1982) Local Hamiltonian Problem for general 𝑙 -local Hamiltonians is QMA -complete for any constant 𝑙 ≥ 2 Kitaev, Kempe, Regev (2006); QMA -complete for the 2D geometry Oliveira and Terhal (2008) Local Hamiltonian Problem for 𝑙 -local stoquastic Hamiltonians is StoqMA -complete for any constant 𝑙 ≥ 2 SB, DiVincenzo, Oliveira, Terhal (2007) Result 2: Local Hamiltonian Problem for TIM on degree-3 graphs is StoqMA- complete.

Implications for Cubitt-Montanaro complexity classification of 2-local Hamiltonians (arxiv:1311.3161): 𝑇 - LHP : special case of the 2-Local Hamiltonian Problem. All terms in the Hamiltonian must belong to some fixed set 𝑇 (with arbitrary real coefficients). Example: 𝑇 = { 𝑎⨂𝑎, 𝑎⨂𝐽, 𝑌⨂𝐽} describes TIM-LHP