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

QIP Sydney January 16, 2015 Based on arXiv: 1410.0703 arXiv: 1402.2295

Quantum Hamiltonian Complexity

Cubitt & Montanaro, arxiv:1311.3161

P NP QMA TIM

Quantum annealing with >100 qubits

Boixo et al, Nature Phys. 10, 218 (2014)

Motivation

Basic model of phase transitions Onsager (1944) Attempts to solve hard optimization problems such as QUBO Understand computational hardness of estimating the ground state energy for quantum spin Hamiltonians

Transverse Ising Model (TIM)

𝐼 =

𝑣

𝑕𝑣 𝑎𝑣 + ℎ𝑣 𝑌𝑣 −

(𝑣,𝑤)

𝐾𝑣,𝑤 𝑎𝑣𝑎𝑤

• Qubits live at vertices of a graph
• Ising 𝑎𝑎 interactions between

nearest neighbor qubits.

• Local magnetic fields along

𝑌 and 𝑎 axes. 𝑎𝑣 = 1 −1 𝑌𝑣 = 0 1 1

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) Easy Hard 𝑗 𝜖| Ψ(𝑢) 𝜖𝑢 = 𝐼(𝑢/𝑈)| Ψ(𝑢)

Unitary evolution

0 ≤ 𝑢 ≤ 𝑈

Easy: 𝐼 0 = − 𝑣 𝑌𝑣 Hard: 𝐼 1 = (𝑣,𝑤) 𝐾𝑣,𝑤 𝑎𝑣𝑎𝑤 + 𝑣 𝑕𝑣𝑎𝑣 𝐼(𝑡) interpolates between 𝐼 0 and 𝐼(1)

Adiabatic Theorem Here 𝜀 is the minimum spectral gap above the ground state of 𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1.

𝑈~ ∥ 𝐼 ∥ 𝜀2 + ∥ 𝐼 ∥2 𝜀3 + ∥ 𝐼 ∥ 𝜀2

Given an adiabatic path 𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1, how large the evolution time 𝑈 should be ? We need a smooth path with a non-negligible spectral gap

Jansen, Seiler, Ruskai, JMP 48, 102111 (2007)

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) ?

𝑁′ 𝑁

Simpler question: can one QA machine efficiently simulate another QA machine ? target QA machine simulator QA machine

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

𝑁′ 𝑁

Simpler question: can one QA machine efficiently simulate another QA machine ?

TIM Hamiltonians

target QA machine simulator QA machine

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

𝑁′ 𝑁

Simpler question: can one QA machine efficiently simulate another QA machine ?

TIM Hamiltonians

Some fixed target class

• f Hamiltonians

target QA machine simulator QA machine

Target Simulator

Adiabatic path

𝐼 𝑡 , 0 ≤ 𝑡 ≤ 1 𝐼′ 𝑡 , 0 ≤ 𝑡 ≤ 1

Number of qubits

𝑜 𝑜′ ≤ 𝑞𝑝𝑚𝑧(𝑜)

Minimum spectral gap

𝜀 𝜀′ ≥ 𝜀

Maximum interaction strength

𝐾 𝐾′ ≤ 𝑞𝑝𝑚𝑧(𝑜, 𝜀−1, 𝐾)

Ground state at 𝑡 = 0

All spins | + All spins | +

Ground state at 𝑡 = 1

| 𝜔 ≈ 𝑊| 𝜔 What does efficient simulation mean ? Here 𝑊: 𝐃2 ⨂𝑜 → 𝐃2 ⨂𝑜′ is a sufficiently simple encoding

𝑁′

TIM Hamiltonians

2-local Hamiltonians target QA machine simulator QA machine

𝑁

When efficient simulation is unlikely:

𝑁′

TIM Hamiltonians

2-local Hamiltonians target QA machine simulator QA machine

𝑁

When efficient simulation is unlikely:

BQP

Aharonov et al (2007) Oliveira and Terhal (2008)

𝑁′

TIM Hamiltonians

2-local Hamiltonians target QA machine simulator QA machine

𝑁

When efficient simulation is unlikely:

BQP

BQP⋂postBPP

Aharonov et al (2007) Oliveira and Terhal (2008) SB, DiVincenzo, Oliveira, Terhal (2007)

𝑁′

TIM Hamiltonians

2-local Hamiltonians target QA machine simulator QA machine

𝑁

When efficient simulation is unlikely:

BQP

BQP⋂postBPP

Aharonov et al (2007) Oliveira and Terhal (2008)

≠

SB, DiVincenzo, Oliveira, Terhal (2007)

More Powerful

Stoquastic k-local Hamiltonians 𝐼 =

𝛽

𝐼𝛽 𝑦|𝐼𝛽|𝑧 ≤ 0 for all 𝑦 ≠ 𝑧 ∈ 0,1 𝑙

System of 𝑜 qubits with a Hamiltonian

• 1. Matrix elements of 𝐼𝛽 in the standard basis are real.
• 2. Off-diagonal matrix elements of 𝐼𝛽 are non-positive:

Each term 𝐼𝛽 acts on at most 𝑙 = 𝑃(1) qubits

Building blocks for 2-local stoquastic Hamiltonians: ±𝑎𝑣, ±𝑎𝑣𝑎𝑤 Diagonal : Elementary interactions: −𝑌 ⊗ 0 0 , −𝑌 ⊗ 1 1 Transverse field: −𝑌𝑣 − 𝑌 ⊗ 𝑌 − 𝑍 ⊗ 𝑍, −𝑌 ⊗ 𝑌 + 𝑍 ⊗ 𝑍

Result 1: universality of TIM for quantum

annealing with 2-local stoquastic Hamiltonians

𝑁′

TIM Hamiltonians

Stoquastic 2-local Hamiltonians target QA machine simulator QA machine

𝑁 =

Result 1: universality of TIM for quantum

annealing with 2-local stoquastic Hamiltonians

𝑁′

TIM Hamiltonians

Stoquastic 2-local Hamiltonians target QA machine simulator QA machine

𝑁 =

with k-local diagonal terms

Result 1: universality of TIM for quantum

annealing with 2-local stoquastic Hamiltonians

𝑁′

TIM Hamiltonians

Stoquastic 2-local Hamiltonians target QA machine simulator QA machine

𝑁 =

with k-local diagonal terms

• n degree-3 graphs

Part II

Computational hardness of estimating the ground state energy of TIM

Local Hamiltonian Problem (LHP): Input: (𝑜, 𝐼 = 𝛽 𝐼𝛽, 𝐷𝑧𝑓𝑡 < 𝐷𝑜𝑝)

𝐹0 = min 𝜔 𝐼 𝜔

Yes-instance: 𝐹0 ≤ 𝐷𝑧𝑓𝑡 No-instance: 𝐹0 ≥ 𝐷𝑜𝑝 Promise: 𝐹0 ∉ 𝐷𝑧𝑓𝑡, 𝐷𝑜𝑝 Normalization: 𝐼𝛽 ≤ 𝑞𝑝𝑚𝑧 𝑜 , 𝐷𝑜𝑝 − 𝐷𝑧𝑓𝑡 ≥ 𝑞𝑝𝑚𝑧 1/𝑜

#terms ≤ 𝑞𝑝𝑚𝑧(𝑜)

Decide which one is the case. Ground state energy:

Merlin-Arthur games (Babai 1985)

Merlin

Unlimited computational power

Arthur

Polynomial-time classical computer

I

instance of yes/no problem

P

proof

accept reject

NP

yes-instance: Arthur accepts some Merlin’s proof no-instance: Arthur rejects any Merlin’s proof

A problem belongs to this class if … complexity class

NP QMA

yes-instance: Arthur accepts some Merlin’s proof no-instance: Arthur rejects any Merlin’s proof yes-instance: Arthur accepts some Merlin’s proof with high probability no-instance: Arthur rejects any Merlin’s proof with high probability Arthur is a quantum computer. Merlin’s proof can be a quantum state.

A problem belongs to this class if … complexity class

NP QMA

yes-instance: Arthur accepts some Merlin’s proof no-instance: Arthur rejects any Merlin’s proof yes-instance: Arthur accepts some Merlin’s proof with high probability no-instance: Arthur rejects any Merlin’s proof with high probability Arthur is a quantum computer. Merlin’s proof can be a quantum state.

A problem belongs to this class if … complexity class

StoqMA

Same as QMA but Arthur can apply only reversible classical gates (CNOT, TOFFOLI) and measure some fixed output qubit in the X-basis. Arthur accepts the proof if the measurement

• utcome is ‘+′. Arthur can use |

0 and | + ancillas.

SB, Bessen, Terhal, arXiv:0611021

P NP QMA StoqMA PostBPP MA AM A0PP SBP Π2

• randomized analogue of

MA NP AM=MA + shared randomness SBP

approximate counting classes

A0PP

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)

Result 2: Local Hamiltonian Problem for TIM

• n degree-3 graphs is StoqMA-complete.

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)

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

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

P NP QMA TIM

𝑇-LHP

contained in P

NP-complete QMA-complete

reducible to TIM-LHP Cubitt-Montanaro (2013):

Example: 𝑇 = { 𝑎⨂𝑎, 𝑎⨂𝐽, 𝑌⨂𝐽} describes TIM-LHP

𝑇-LHP

contained in P

NP-complete QMA-complete StoqMA-complete

Improved Cubitt-Montanaro:

StoqMA P NP QMA StoqMA

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).

Part III

Ferromagnetic TIM 𝐼 =

𝑣

𝑕 𝑎𝑣 + ℎ𝑣 𝑌𝑣 −

(𝑣,𝑤)

𝐾𝑣,𝑤 𝑎𝑣𝑎𝑤

𝐾𝑣,𝑤 ≥ 0

Uniform Z-field

Classical ferromagnetic Ising model: known results

Uniform Z-field: trivial: ↑ ↑ ↑ ↑ ↑ ↑ ↑ or ↓ ↓ ↓ ↓ ↓ ↓ ↓ Arbitrary Z-fields: 𝑃(𝑜3) algorithm (equivalent to Min Cut problem) Computing the minimum energy:

Classical ferromagnetic Ising model: known results

Uniform Z-field: trivial: ↑ ↑ ↑ ↑ ↑ ↑ ↑ or ↓ ↓ ↓ ↓ ↓ ↓ ↓ Arbitrary Z-fields: 𝑃(𝑜3) algorithm (equivalent to Min Cut problem) Computing the minimum energy: Computing the partition function Tr 𝑓−𝐼 : Exact computation is #𝑄-hard, Jerrum & Sinclair (1993) Uniform Z-field: 𝑃(𝑜17𝜀−2) approximation algorithm Jerrum & Sinclair (1993) Arbitrary Z-fields: approximation is #𝐶𝐽𝑇-hard. Unlikely to have poly-time algorithm, Goldberg & Jerrum (2005)

𝑎 = Tr 𝑓−𝐼

Result 3: Polynomial-time approximation algorithm for

the partition function of the ferromagnetic TIM.

𝑎 = Tr 𝑓−𝐼

Classical randomized algorithm

𝐾𝑣,𝑤 𝑕, ℎ𝑣

𝜀

𝑎

1 − 𝜀 𝑎 ≤ 𝑎 ≤ 1 + 𝜀 𝑎 𝑃 𝑜59𝐾21𝜀−9

running time

𝐾 = max 𝐾𝑣,𝑤, |ℎ𝑣|, |𝑕| 𝑜 = number of spins w.h.p.

Result 3: Polynomial-time approximation algorithm for

the partition function of the ferromagnetic TIM.

𝑎 = Tr 𝑓−𝐼/𝑈

• 1. The free energy 𝐺 𝑈 = −𝑈log 𝑎

can be estimated with an additive error 𝜀 in time 𝑞𝑝𝑚𝑧(𝑜, 𝜀−1, 𝐾𝑈−1)

Result 3: Polynomial-time approximation algorithm for

the partition function of the ferromagnetic TIM. Implications:

𝑎 = Tr 𝑓−𝐼/𝑈

• 1. The free energy 𝐺 𝑈 = −𝑈log 𝑎

can be estimated with an additive error 𝜀 in time 𝑞𝑝𝑚𝑧(𝑜, 𝜀−1, 𝐾𝑈−1)

• 2. The ground state energy 𝐹0 can be estimated with an

additive error 𝜀 in time 𝑞𝑝𝑚𝑧(𝑜, 𝜀−1, 𝐾)

Result 3: Polynomial-time approximation algorithm for

the partition function of the ferromagnetic TIM. Implications:

Sketch of the proofs

Ferromagnetic TIM is easy

𝐼 = −𝐵 − 𝐶 𝐵 = classical ferromag.

Ising model

𝐶 = transverse field 𝑎 = Tr 𝑓𝐵+𝐶

Ferromagnetic TIM is easy

𝑎′ = Tr 𝑓𝐵/𝑠𝑓𝐶/𝑠 𝑠

𝑠 = 𝑞𝑝𝑚𝑧(𝑜)

𝐼 = −𝐵 − 𝐶 𝐵 = classical ferromag.

Ising model

𝐶 = transverse field 𝑎 = Tr 𝑓𝐵+𝐶

Trotter-Suzuki approximation to 𝑎

Ferromagnetic TIM is easy

𝑎′ = Tr 𝑓𝐵/𝑠𝑓𝐶/𝑠 𝑠

𝑠 = 𝑞𝑝𝑚𝑧(𝑜)

Fact 1: 𝑎′ approximates 𝑎 with a multiplicative error 𝑃(𝜀) if

𝐼 = −𝐵 − 𝐶 𝐵 = classical ferromag.

Ising model

𝐶 = transverse field 𝑎 = Tr 𝑓𝐵+𝐶

Trotter-Suzuki approximation to 𝑎

𝑠 ≥ 𝜀−1/2 𝐵 3/2 + 𝐶 3/2

Ferromagnetic TIM is easy

𝑎′ = Tr 𝑓𝐵/𝑠𝑓𝐶/𝑠 𝑠

𝑠 = 𝑞𝑝𝑚𝑧(𝑜)

Fact 1: 𝑎′ approximates 𝑎 with a multiplicative error 𝑃(𝜀) if

𝐼 = −𝐵 − 𝐶 𝐵 = classical ferromag.

Ising model

𝐶 = transverse field 𝑎 = Tr 𝑓𝐵+𝐶

Trotter-Suzuki approximation to 𝑎 Fact 2: (Quantum-to-Classical mapping) 𝑎′ coincides with the partition function of a classical ferromagnetic Ising model with 𝑜′ = 𝑜𝑠 spins.

𝑠 ≥ 𝜀−1/2 𝐵 3/2 + 𝐶 3/2

Ferromagnetic TIM is easy

𝑎′ = Tr 𝑓𝐵/𝑠𝑓𝐶/𝑠 𝑠

𝑠 = 𝑞𝑝𝑚𝑧(𝑜)

Fact 1: 𝑎′ approximates 𝑎 with a multiplicative error 𝑃(𝜀) if

𝐼 = −𝐵 − 𝐶 𝐵 = classical ferromag.

Ising model

𝐶 = transverse field 𝑎 = Tr 𝑓𝐵+𝐶

Trotter-Suzuki approximation to 𝑎 Fact 2: (Quantum-to-Classical mapping) 𝑎′ coincides with the partition function of a classical ferromagnetic Ising model with 𝑜′ = 𝑜𝑠 spins. Fact 3: [Jerrum & Sinclair 1993] The partition function of the classical ferromagnetic Ising model can be approximated in time 𝑃 𝑜17𝜀−2 by a Monte Carlo algorithm.

𝑠 ≥ 𝜀−1/2 𝐵 3/2 + 𝐶 3/2

Sketch of the proofs

(part I and II)

perturbative reductions Kitaev, Kempe, Regev (2004)

𝐼′ = 𝐼0 + 𝑊

𝐼

target Hamiltonian simulator Hamiltonian

𝐼 ≈ 𝐼eff = 𝑊

−− − ∆−1 𝑊 −+ 𝑊 +− + ⋯

effective low-energy Hamiltonian 𝐹0 𝐹0 + Δ

𝑊

−+

𝑊

+−

degenerate ground subspace of 𝐼0

𝑊

−−

TIM on degree-3 graphs

Stoquastic 2-local Hamiltonians energy

Perturbative reductions

simulator target

General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy

Perturbative reductions

simulator target

simulator General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy target

Perturbative reductions

simulator General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy target

Perturbative reductions

simulator General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy target

Perturbative reductions

simulator General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy target

Perturbative reductions

simulator General TIM

TIM on degree-3 graphs

Hard-core dimers Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy target

Perturbative reductions

Hard-core dimers model (HCD)

• System of 𝑙 particles on a fixed graph with 𝑜 nodes.
• Each site can be either empty or occupied by a particle
• Admissible configurations are nearest-neighbor pairs - dimers
• Dimers must be separated by a fixed distance 𝑠 – the range

1 2 3 4 5 6 7 8 9

range-2 HCD

Hard-core dimers model (HCD) 𝐼 = −𝑢

𝑣,𝑤

𝑋

𝑣,𝑤

+

𝑣

𝜈𝑣𝑂𝑣 +

𝑣,𝑤

𝐾𝑣,𝑤𝑂𝑣𝑂𝑤

long-range hopping

• n-site chemical

potential two-particle interaction

1 2 3 4 5 6 7 8 9

range-2 HCD

• System of 𝑙 particles on a fixed graph with 𝑜 nodes.
• Each site can be either empty or occupied by a particle
• Admissible configurations are nearest-neighbor pairs - dimers
• Dimers must be separated by a fixed distance 𝑠 – the range

𝑂3 = 0 𝑂1 = 1

Dimers can only move locally: 𝐼 = −𝑢

𝑣,𝑤

𝑋

𝑣,𝑤

+

𝑣

𝜈𝑣𝑂𝑣 +

𝑣,𝑤

𝐾𝑣,𝑤𝑂𝑣𝑂𝑤

long-range hopping

• n-site chemical

potential two-particle interaction

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

𝑋

4,6

Allowed hopping:

range-2 HCD

𝐼 = −𝑢

𝑣,𝑤

𝑋

𝑣,𝑤

+

𝑣

𝜈𝑣𝑂𝑣 +

𝑣,𝑤

𝐾𝑣,𝑤𝑂𝑣𝑂𝑤

long-range hopping

• n-site chemical

potential two-particle interaction

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

range-2 HCD

Forbidden hopping:

𝑋

3,6

Dimers can only move locally:

𝐼 = −𝑢

𝑣,𝑤

𝑋

𝑣,𝑤

+

𝑣

𝜈𝑣𝑂𝑣 +

𝑣,𝑤

𝐾𝑣,𝑤𝑂𝑣𝑂𝑤

long-range hopping

• n-site chemical

potential two-particle interaction

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

range-2 HCD

Forbidden hopping:

𝑋

5,7

Dimers cannot come too close to each other:

How the reductions work: overview

General TIM

TIM on degree-3 graphs

Hard-core dimers (range-3) Hard-core bosons (range-2) Hard-core bosons (range-1) Hard-core bosons w. controlled hopping Stoquastic 2-local Hamiltonians energy

TIM on degree-3 graphs

5 1 2 3 4

Encode each spin into the ground subspace of 1D TIM. Now each spin is coupled to at most 3 other spins.

General TIM 1 2 3 4 5 1 2 3 4

General TIM Hard-core dimers (range-3)

General TIM Hard-core dimers (range-3)

Ising Hamiltonian whose ground states are range-3 dimers:

𝐼0 =

𝑣

𝑂𝑣 − 2

𝐸 𝑣,𝑤 =1

𝑂𝑣𝑂𝑤 + Γ

𝐸 𝑣,𝑤 =2

𝑂𝑣𝑂𝑤 Γ = 𝑞𝑝𝑚𝑧(𝑜) 𝑂𝑣 = (𝐽 + 𝑎𝑣)/2

𝐸(𝑣, 𝑤) – graph distance between sites 𝑣, 𝑤

General TIM Hard-core dimers (range-3) Hopping

𝑊

+−

𝑊

−+

𝑊 = ℎ

𝑣

𝑌𝑣

The intermediate state created by 𝑊

“remembers” the dimer location.

This is why local hopping can emerge from the global transverse field and this is why we need dimers.

Open problems:

Universality of TIM for quantum annealing with 𝑙-local stoquastic Hamiltonians for 𝑙 > 2 Is there a subclass of BQP that captures the power

• f quantum annealing with stoquastic Hamiltonians ?

More efficient algorithms for the ferromagnetic TIM. Can one compute the ground state energy directly without computing the partition function ? Amplification of the completeness and soundness errors for the class StoqMA