digital quantum simulation - uni stuttgart€¦ · plaquette interaction (light red) a ... g r y y...

Digital Quantum Simulation

Hendrik Weimer

Institute for Theoretical Physics, Leibniz University Hannover

Blaubeuren, 22 July 2014

Leibniz University Hannover

Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation

Course on Quantum Simulation

Leibniz Universitat Hannover, Summer 2013

Lecture notes:

http://v.gd/qsim2013

Wikiversity page:

http://en.wikiversity.org/wiki/Quantum Simulation


Outline

1 Why quantum simulation?

2 Simulation of coherent dynamics

3 Dissipative quantum state engineering


Strongly correlated systems

Strongly correlated systemsare difficult to describetheoretically

Exponentially growingHilbert space dimension

High-temperaturesuperconductors

Quark bound states(protons, neutrons)

Frustrated quantum magnets


Exact diagonalization

Slightly misleading: only as “exact” as your computer’s arithmeticprecision

Time-independent Schrodinger equation

H|ψ〉 = E|ψ〉

Ground state: find |ψ〉 such that 〈ψ|H|ψ〉 is minimal

Example: Transverse field Ising chain

H = g∑i

σ(i)x −

∑i

σ(i)z σ(i+1)

z

Exponential complexity: dimH = 2N

N = 40: 8 TB of memoryN = 300: more basis states than atoms in the universe!


Quantum Monte-Carlo

Quantum-classical mapping for the partition function

Z = Tr {exp(−βH)} = Tr

{exp

[−β(g∑i

σ(i)x −

∑i

σ(i)z σ(i+1)

z

)]}

Suzuki-Trotter formula:

exp

[1

N(A+B)

]= exp

(A

N

)exp

(B

N

)+O(1/N2)

Z = limNy→∞

Tr

exp

(− βgNy

∑i

σ(i)x

)Ny

exp

(β

Ny

∑i

σ(i)z σ(i+1)

z

)Ny

Ny multiplications: additional dimension


Quantum Monte-Carlo II

Classical partition function

Z = ΛNNyTr

exp

γ N∑i=1

Ny∑j=1

σ(i,j)z σ(i,j+1)

z +β

Ny

N∑i=1

Ny∑j=1

σ(i,j)z σ(i+1,j)

z

Classical temperature βcl = β/Ny 6= β

Solve using standard Monte Carlo methods (Metropolis algorithm)N. Metropolis et. al., J. Chem. Phys. 21, 1087 (1953)

Quantum-classical mapping does not always work

H = J∑〈ij〉

σ(i)+ σ

(i)− + H.c.

Antiferromagnetic interaction on a non-bipartite lattice: negativeprobabilities in the corresponding classical model (sign problem)


Density-Matrix Renormalization Group

Matrix product state (MPS) ansatzU. Schollwock, Ann. Phys. 326, 92 (2011)

|ψ〉 =∑i

Tr

N∏j=1

Aj

|i〉A is a D ×D-dimensional matrix (D � dimH)How much information does an MPS contain? Entanglement entropy

S = −Tr {ρA log ρA} ≤ 2 logD

Area law of entanglement entropyJ. Eisert et al., Rev. Mod. Phys. 82, 277 (2010)

S(ρA) . A(A)

MPS are good only for one-dimensional systems with short-rangedinteractions


Universal quantum simulator

Simulating quantum mechanics with otherquantum systemsR. P. Feynman, Int. J. Theo. Phys. 21, 467 (1982)

Universal Quantum Simulator (UQS):device simulating the dynamics of anyother quantum system with short-rangeinteractionsS. Lloyd, Science 273, 1073 (1996)

Digital quantum simulator: UQS fordiscrete time steps


Analog vs digital

HW, M. Muller, H. P. Buchler, I. Lesanovsky, Quant. Inf. Proc. 10, 885 (2010)

Analog quantum simulators have dominant two-body interactionsStrength of three-, four-, five-body interactions decays exponentiallyTurning off two-body interactions requires enormous fine-tuningDigital quantum simulator is non-perturbative and does not requirefine-tuning


Outline





The Toric Code

Control spins (red) and ensemble spins(blue) on a 2D lattice

Plaquette interaction (light red)

Ap = σ(i)x σ(j)

x σ(k)x σ(l)

x

Site interaction (green)

Bs = σ(i)z σ(j)

z σ(k)z σ(l)

z

Control spins only mediate the interactionsToric code HamiltonianA. Kitaev, Ann. Phys. 303, 2 (2003)

H = −E0

(∑p

Ap +∑s

Bs

)Each Ap and Bs has to eigenvalues ±1 (eightfold degenerate)All Ap and Bs commute: ground state has Ap = 1 and Bs = 1


From N + 1-body gates to N -body interactions

Goal: simulate an effective plaquette interactionH = Ap = σ

(i)x σ

(j)x σ

(k)x σ

(l)x

Implement the time dependent Schrodingerequation

i~d

dt|ψ〉 = H|ψ〉

Time-independent Hamiltonian

|ψ(t)〉 = exp(−iHt/~)|ψ(0)〉

Here: implement U = exp(−iHt/~) using singlequbit gates plus Controlled-NOTN

UCNOTN = |0〉〈0|cN⊗i=1

1i + |1〉〈1|cN⊗i=1

σxi

sj

sj

s

j...

..

.


Mapping onto control atom

Map |±1〉 eigenstates of Ap onto |0〉, |1〉 of the control atom

Ry = exp(−iπσy/4) =1√2

(1 −11 1

)

|0〉 |0〉|0〉 |1〉

|+〉|+〉|−〉|−〉

G

Ry R†y

CN

OTN


Coherent dynamics

Apply the mapping G transferring the eigenvalue of Ap onto thecontrol spin

Write a phase exp(−iφσz) onto the control spin

Undo the mapping G = G−1

|0〉 |0〉

G G†

e−iφσz

e−iHτ/~

Reset

Simulates the Hamiltonian H at discrete times t = kτ (digital)

Energy scale E0 = ~φ/τHW, M. Muller, I. Lesanovsky, P. Zoller, H. P. Buchler, Nature Phys. 6, 382 (2010)


Scaling up to the full lattice model

Implement Bs = σ(i)z σ

(j)z σ

(k)z σ

(l)z the same way as Ap by swapping

σx for σz using local gatesTotal Hamiltonian

H = −E0

(∑p

Ap +∑s

Bs

)=∑λ

hλ

Suzuki-Trotter decomposition

U(τ) = exp(−iHτ/~) =∏λ

exp(−hλτ/~) +O(τ2)

Straightforward parallelimplementation

But: only one hλ acting oneach spin at a time

HW, Mol. Phys. 111, 1753 (2013)


Robustness against errors

Trotterization errors due to non-commuting interactions

Errors in many-body gate

Ug = |0〉〈0|c ⊗ eiφQ + |1〉〈1|c ⊗Ap

Modified Hamiltonian h = −(Ap +Q), additional dephasing

ρ→ ρ− iφ [h, ρ]− φ2

2[h, [h, ρ]] +

φ2

2

(2QρQ−

{Q2, ρ

})Tolerable if quantum phase stable against fluctuations (no errorcorrection required)

Dephasing leads to an effective heating


Outline





Quantum state engineering

Tailored quantum states are an importantresource (quantum simulation, quantumcommunication, quantum metrology, ...)

Previously: coherent evolution (adiabaticfollowing, quantum logic gates)

New tool: controlled dissipationS. Diehl et al., Nature Phys. 4, 878 (2008)

F. Verstraete et al., Nature Phys. 5, 633 (2009)

HW et al., Nature Phys. 6, 382 (2010)

Engineer a suitable attractor state of thedynamics

Inherently more robust

Position

Vel

ocity


Quantum master equation

Open quantum system described by a quantum master equation(Lindblad form)

dρ

dt= −i~ [H, ρ] +

∑n

γn

(cnρc

†n −

1

2{c†ncn, ρ}

)

ρ =∑i

pi|ψi〉〈ψi| Density operator

H Hamiltonian

cn Quantum jump operators (non-Hermitian)

γn Decay rate


Stationary state

Sufficient (and usually also necessary) condition: dρdt = 0

Special case: pure state ρ = |ψ〉〈ψ|More general: von Neumann entropy

S = −Tr {ρ log ρ}Only Hermitian jump operators⇒ Maximally mixed state (S = log d)

ρ =

1/d1/d

. . .

dρ

dt= −i~ [H, ρ] +

∑n

γn

(cnρc

†n −

1

2{c†ncn, ρ}

)


Dissipative cooling

Goal: cool into the ground state of H = Ap (−1 eigenstate)

Use the same mapping (ensemble 7→ control) as before

Instead of writing a phase on the control spin: controlled spin flip ofone random ensemble spin j

U = |0〉〈0|c ⊗ 1 + |1〉〈1|c ⊗ exp(iφσ(j)z )

If we do a spin flip: control atom will not end in |0〉Reset spin (incoherent) from |1〉 to |0〉Discrete Markovian master equation

ρ(t+ τ) = ρ(t) + γ

(cρc† − 1

2

{c†c, ρ

})Rate γ = φ2/τ , jump operator c = σ

(j)z (1 +Ap)/2


Scaling up

Cooling: random walk of the anyons

Averaging over 103 realizations of the dynamics

Imperfections: residual anyon density n

0

0.1

0.2

0.3

0.4

0 20 40 60t[γ−1]

n


Linear response theory

Gate error probability ε: probability to endup in a state orthogonal to the desired one

Toric code: gate errors create anyons

0

0.4

0.8

1.2

1.6

0 0.01 0.02 0.03 0.04 0.05

T[E

0/kB]

ε

0

0.2

0.4

0.6

0 2 4 6 8 10

n

t[h/E0]

HW, Mol. Phys. 111, 1753 (2013)

Uncorrelated errors⇒Effective temperature

T ≈ − 2E0

kB log n

Anyon density n within linearresponse: n = 14ε

⇒ Effective temperaturebenchmarks the quantumsimulator


Experimental realization

Proof of principle experiment with trapped ions

N -body Mølmer-Sørensen gateA. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971 (1999)

Four ensemble spins + 1 control ion

Minimal instance of a toric code Hamiltonian (1 plaquette)

J. Barreiro et al., Nature 470, 486 (2011)


Herrenhausen Castle


Herrenhausen Castle

Funded by:


PhD and Postdoc Positions Available!

Freigeist project “Quantum States on Demand”

Quantum state engineering

Dissipative many-body quantum dynamics


H = U∑o

(∑i∈o σ

(i)z

)2− J∑

pBp + V

∑pB2p

Ring-exchange Bp = σ+σ−σ+σ− + h.c via gate sequence

Low-energy sector (U � J, V ): three spins up/down on eachoctahedron

V = J : Rokhsar-Kivelson point (non-stabilizer state)

V < J : Spin liquid phase with Coulombic 1/r interactions


2D Fermi-Hubbard model isbelieved to be realized inhigh-temperature cupratesuperconductors

H = −t∑<ij>σ

c†iσcjσ +U∑i

ni↑ni↓

Mapping fermions onto spins:Jordan-Wigner transformation

Problem in 2D: Wigner strings(highly nonlocal interactions)

Solution: Introduce auxillaryfermion fieldVerstrate, Cirac, J. Stat. Mech. 2005, P09012

(2005)

1 1’ 2 2’ 3 3’ 4 4’

5 5’6 6’7 7’8 8’

9 9’ 10 10’ 11 11’ 12 12’

13 13’14 14’15 15’16’16

HW, M. Muller, H. P. Buchler, I.

Lesanovsky, Quant. Inf. Proc. 10, 885

(2010)

Haux = −V∑{i,j}σ

Pi′,j′Pj′+1,i′−1

Pi′,j′ = (di′σ+d†i′σ)(dj′σ−d†j′σ)

Results in localsix-body interactions


digital quantum simulation - uni stuttgart€¦ · plaquette interaction (light red) a ... g r y y...

Documents