Quantum Simulations with Superconducting Qubits

Nikolaj Moll, Daniel Egger, Stefan Filipp, Andreas Fuhrer, Marc Ganzhorn, Peter Müller, Marco Roth, and Sebastian Schmidt

Quantum Simulations with

Superconducting Qubits

Quantum Technology

Team members

§ Panagiotis Barkoutsos

§ Daniel Egger

§ Stefan Filipp

§ Andreas Fuhrer

§ Marc Ganzhorn

§ Andreas Kuhlmann

§ Nikolaj Moll

§ Peter Müller

§ Walter Riess

§ Marco Roth

§ Peter Staar

Collaborators

§ IBM Yorktown Quantum Technology Team

§ David DiVincenzo, RWTH Aachen

§ Wolfgang Lechner, IQOQI

§ Sebastian Schmidt, ETHZ

§ Matthias Troyer, ETHZ

§ Andreas Wallraff, ETHZ

§ Martin Weides and Michael Marthaler, KIT

§ Frank Wilhelm-Mauch, Saarland University

Outline

§ Motivation

§ Quantum chemistry

– Digital vs analog

– Adiabatic quantum simulation

§ Experimental realization adiabatic quantum simulation

– Tunable coupler

§ Outlook

3

Quantum simulations

Exponential resources to store wave-function (2^" complex coefficients)

Solution: Quantum simulator. Systems with equivalent dynamics, which can be well controlled and measured. [Feynman, 1982; Lloyd, 1996]

Applications: Quantum chemistry, Quantum systems (high-Tc, spin glasses)

Qubits Memory Time for one gate

10 16 kB microseconds on a watch

20 16 MB milliseconds on a smartphone

30 16 GB seconds on a laptop

40 16 TB minutes on a supercomputer

50 16 PB days on a top supercomputer

60 16 EB long long time

80 size of the visible universe age of the universe from Troyer

Outline

§ Motivation

§ Quantum chemistry

– Digital vs analog

– Adiabatic quantum simulation

§ Experimental realization adiabatic quantum simulation

– Tunable coupler

§ Outlook

5

Quantum chemistry

Schrödinger equation in the Born-Oppenheimer approximation 𝐸(𝑅) Ψ 𝑟, 𝑅 = −^-_.∑𝛻^. + ∑ ₅²³²⁴

365₄ + ∑ ₅²⁷

768₉ + ∑_|8 ^-

;68₉| Ψ 𝑟

full configuration interaction (full CI) for the hydrogen molecule: exact solution within the basis basis atomic orbitals 𝜑_=-and 𝜑_=- give two molecular orbitals

𝜑_> = 𝜑_=- + 𝜑_=. and 𝜑_? = 𝜑_=- − 𝜑_=.

gives the two electron wave function

Ψ r_-, r_. = c_- 𝜑_> 𝑟_- 𝜑_? 𝑟_. − 𝜑_? 𝑟_- 𝜑_> 𝑟_. + 𝑐_. 𝜑_> 𝑟_- 𝜑_? 𝑟_. + 𝜑_? 𝑟_- 𝜑_> 𝑟_. the coefficients 𝑐_-and 𝑐_. have to be determined 𝐸 = min^{〈G = G〉}_〈G|G〉

𝜑_=- 𝜑_=.

Quantum chemistry

7

Simulating Chemistry using Quantum Computers 25

Second-quantized First-quantized

Wavefunction encoding

Fock state in a given basis:

| i=|0100i

On a grid of 2ⁿsites per dimension:

| i=X

x

ax|xi

Qubits required to represent the wavefunction

One per basis state (spin-orbital) 3nper particle (nuclei & electrons)

Molecular Hamiltonian

X

pq

hpqa^†_paq+1 2

X

pqrs

hpqrsa^†_pa^†_qaras

Coefficients pre-computed classically

X

i

p²_i

2m_i +X

i<j

q_iq_j r_ij

Interaction calculated on the fly Quantum gates

required for sim- ulation

O(M⁵) with number of basis states O(B²) with number of particles

Advantages

• Compact wavefunction repre- sentation (requires fewer qubits)

• Takes advantage of classical electronic-structure theory to improve performance

• Already experimentally imple- mented

• Better asymptotic scaling (re- quires fewer gates)

• Treats dynamics better

• Can be used for computing reac- tion rates or state-to-state tran- sition amplitudes

Table 1: Comparison of second- and first-quantization approaches to quantum simulation.

Simulating Chemistry using Quantum Computers 25

Second-quantized First-quantized

Wavefunction encoding

Fock state in a given basis:

| i=|0100i

On a grid of 2ⁿsites per dimension:

| i=X

x

ax|xi

Qubits required to represent the wavefunction

One per basis state (spin-orbital) 3nper particle (nuclei & electrons)

Molecular Hamiltonian

X

pq

hpqa^†_paq+1 2

X

pqrs

hpqrsa^†_pa^†_qaras

Coefficients pre-computed classically

X

i

p²_i

2m_i +X

i<j

q_iq_j r_ij

Interaction calculated on the fly Quantum gates

required for sim- ulation

O(M⁵) with number of basis states O(B²) with number of particles

Advantages

• Compact wavefunction repre- sentation (requires fewer qubits)

• Takes advantage of classical electronic-structure theory to improve performance

• Already experimentally imple- mented

• Better asymptotic scaling (re- quires fewer gates)

• Treats dynamics better

• Can be used for computing reac- tion rates or state-to-state tran- sition amplitudes

Table 1: Comparison of second- and first-quantization approaches to quantum simulation.

I. Kassal, J.D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Ann. Rev. Phys. Chem. 62, 185 (2011).

Second quantization on a quantum computer

Hamiltonian in second quantization

𝐻 = ∑ℎ_KL𝑎_K^N𝑎_L + ^-_.∑ℎ_KL8O𝑎_K^N𝑎_L^N𝑎₈𝑎_O number of terms ~ 𝑁^R where

ℎ_KL = ∫ 𝜙_K^∗ 𝑟 −^-_.𝛻^. − ∑₅²⁷

768 𝜙_L 𝑟 𝑑𝑟 and ℎ_KL8O = ∫ ^{WX∗ 8Y WZ∗} ₈^{8[ W\ 8[ W] 8Y}

Y68_[ 𝑑𝑟_-𝑑𝑟_. replace creation 𝑎_^^N and 𝑎_^ annihilation operator with corresponding Pauli operators

𝑎_^^N → 𝜎_^^a = ^-_. 𝑋 + 𝑖𝑌 = 0 1 0 0 𝑎_^ → 𝜎_^⁶ = ^-_. 𝑋 − 𝑖𝑌 = 0 0

1 0

Problem: Pauli operators fulfill boson statistics 𝜎_^, 𝜎_^ = 0, 𝜎_^^a, 𝜎_^^a = 0, 𝜎_^, 𝜎_^^a = 𝛿_^,h

Jordan-Wigner transformation and hydrogen molecule

Electrons are Fermions and obey

𝑎_^, 𝑎_^ = 0, 𝑎_^^N, 𝑎_^^N = 0, 𝑎_^, 𝑎_^^N = 𝛿_^,h

Jordan-Wigner transformation solves this by accounting for the parity

𝑎_^ = ∏^{^6-}_hk-𝑍_h 𝑋_^ − 𝑖𝑌_^ and 𝑎_^^N= ∏^{^6-}_hk-𝑍_h 𝑋_^ + 𝑖𝑌_^ |00𝑛_h10〉 → ±|01n_n00〉

hydrogen molecule with minimal STO-3G basis has 4 qubits 𝜒_- = 𝜙_- ↑ , 𝜒_. = 𝜙_. ↑ , 𝜒_q = 𝜙_- ↓ , 𝜒_s = 𝜙_. ↓ gives Hamiltonian expressed in Pauli matrices

H_u[ = 𝑓_- + 𝑓_.𝑍_-𝑍_. + 𝑓_.𝑍_q𝑍_s + 𝑓_q𝑍_-𝑍_q + 𝑓_q𝑍_.𝑍_s + 𝑓_s𝑍_.𝑍_q + 𝑓_w𝑍_-𝑍_s

+𝑓_R𝑋_-𝑋_.𝑋_q𝑋_s +𝑓_R𝑋_-𝑋_.𝑌_q𝑌_s + 𝑓_R𝑌_-𝑌_.𝑋_q𝑋_s + 𝑓_R 𝑌_-𝑌_.𝑌_q𝑌_s + 𝑓_x𝑍_- + 𝑓_x𝑍_s + 𝑓_y𝑍_. + 𝑓_y𝑍_q

9 4-local terms

Reduction of the qubits for the hydrogen molecule

Hilbert space is block diagonal with respect to the particle number 𝑁

N. Moll, A. Fuhrer, P. Staar, I. Tavernelli, J. Phys. A: Math. Theor. 49, 295301 (2016).

Reduction of the qubits for the hydrogen molecule

Hilbert space is block diagonal with respect to the particle number 𝑁

Project out all states 𝑁 ≠ 2 (only singlet states):

𝐻_{ = 𝑃^N𝐻𝑃 with 𝑃 = 𝑁_↑ 2 − 𝑁_↑ 𝑁_↓(2 − 𝑁_↓)

11

N. Moll, A. Fuhrer, P. Staar, I. Tavernelli, J. Phys. A: Math. Theor. 49, 295301 (2016).

Reduction of the qubits for the hydrogen molecule

Hilbert space is block diagonal with respect to the particle number 𝑁

Project out all states 𝑁 ≠ 2 (only singlet states):

𝐻_{ = 𝑃^N𝐻𝑃 with 𝑃 = 𝑁_↑ 2 − 𝑁_↑ 𝑁_↓(2 − 𝑁_↓) Shift and copy the blocks.

N. Moll, A. Fuhrer, P. Staar, I. Tavernelli, J. Phys. A: Math. Theor. 49, 295301 (2016).

Reduction of the qubits for the hydrogen molecule

Hilbert space is block diagonal with respect to the particle number 𝑁

Project out all states 𝑁 ≠ 2 (only singlet states):

𝐻_{ = 𝑃^N𝐻𝑃 with 𝑃 = 𝑁_↑ 2 − 𝑁_↑ 𝑁_↓(2 − 𝑁_↓) Shift and copy the blocks.

The Hamiltonian of the hydrogen molecule with two qubits:

𝐻_. = 𝑓_- − 2 𝑓_. + 4 𝑓_R 𝑋_- 𝑋_. + −2 𝑓_q + 𝑓_s + 𝑓_w 𝑍_- 𝑍_. + 𝑓_x − 𝑓_y 𝑍_- + 𝑓_x − 𝑓_y 𝑍_.

13

N. Moll, A. Fuhrer, P. Staar, I. Tavernelli, J. Phys. A: Math. Theor. 49, 295301 (2016).

Digital versus analog quantum computing

In digital quantum computing the Hamiltonian is split into components that are evaluated in sequence using the Trotter decomposition:

𝑒^{^∑=4•} = ∏ 𝑒^{^=4•/• •} + ⋯ For example, the term

𝑒^{6^„ 2Y2[2…}

is evaluated CNOT gates and one𝑅_Wgate

In analog quantum computing the whole Hamiltonian is directly implemented in the quantum computer and evaluated at once.

𝑅

q1 q2 q3

Adiabatic quantum annealing

Adiabatic theorem: a system initialized in an eigenstate of some Hamiltonian H(t) will remain in that eigenstate if the change of the Hamiltonian is sufficiently slow.

Basic idea: 𝐻 𝑡 = 𝐻_‡(1 − 𝑠 𝑡 ) + 𝐻_‰𝑠(𝑡), with 𝑠 𝑡 ∈ 0,1 . easy to initialize difficult many-body problem

Hydrogen Molecule:

𝐻_‡ = 𝛿_-𝑋_- + 𝛿_.𝑋_.

𝐻_‰ = 𝜀_-𝑍_- + 𝜀_.𝑍_. + 𝐽_Ž𝑋_-𝑋_. + 𝐽_•𝑍_-𝑍_.

15

Adiabatic hydrogen simulation

target energy

Instantaneous state 𝜓(𝑡) = 𝛼_-(𝑡) 00 + 𝛼_.(𝑡) 01 + 𝛼_q(𝑡) 10 + 𝛼_s(𝑡) 11 Ground state of 𝐻_‡: 𝛼_- = −𝛼_.= −𝛼_q= 𝛼_s = ^-_.

Ground state of 𝐻_‰: 𝛼_- = 0.11, 𝛼_s = −0.99, 𝛼_. = 𝛼_q = 0

Adiabatic hydrogen simulation

Estimation of annealing duration to achieve chemical accuracy Δ𝐸 ≤ 10^6w𝐻𝑎𝑟𝑡𝑟𝑒𝑒 (0.3 𝑚𝑒𝑉)

17

𝐻_- 𝑅 𝐻_.

Outline

§ Motivation

§ Quantum chemistry

– Digital vs analog

– Adiabatic quantum simulation

§ Experimental realization adiabatic quantum simulation

– Tunable coupler

§ Outlook

Tunable-coupler

Sum gate or bSWAP: modulation frequency 𝜔_š = 𝜔›_- + 𝜔›_.. Difference gate or iSWAP: modulation frequency 𝜔_œ = 𝜔›_- − 𝜔›_.

|01⟩ |10⟩

|11⟩

|00⟩

Difference gate iSWAP Sum gate

bSWAP

𝐻/ℏ = 𝜔›_-

2 𝑍_- +𝜔›_. 2 𝑍_.

+^Ÿ _.^(•) 𝑋_-𝑋_.− 𝑌_-𝑌_. +^Ÿ¡_.^(•) 𝑋_-𝑋_.+ 𝑌_-𝑌_.

iSWAP bSWAP

𝐽_¢,£ 𝑡 = 𝐽_¢,£ 0 + Ω_¢,£cos 𝜔_¢,£𝑡

Q1 Q2 TC

𝜔_L (GHz) 4.981 4.404 6.01 (max)

𝛼 (GHz) -0.366 -0.346 -

𝑇_- (𝜇𝑠) 72 45 25

𝑇_. (𝜇𝑠) 35 35 1-2

Tunable-coupler: time-resolved Rabi spectroscopy

Sum gate or bSWAP: modulation frequency 𝜔_š = 𝜔›_- + 𝜔›_.. Difference gate or iSWAP: modulation frequency 𝜔_œ = 𝜔›_- − 𝜔›_.

|01⟩ |10⟩

|11⟩

|00⟩

Difference gate iSWAP Sum gate

bSWAP

𝐻/ℏ = 𝜔›_-

2 𝑍_- +𝜔›_. 2 𝑍_.

+^Ÿ _.^(•) 𝑋_-𝑋_.− 𝑌_-𝑌_. +^Ÿ¡_.^(•) 𝑋_-𝑋_.+ 𝑌_-𝑌_.

iSWAP bSWAP

𝐽_¢,£ 𝑡 = 𝐽_¢,£ 0 + Ω_¢,£cos 𝜔_¢,£𝑡

iSWAP bSWAP

𝜃 = −0.108 𝜙‡, 𝛿 = 0.05 𝜙‡ 𝜃 = −0.108 𝜙_‡, 𝛿 = 0.12 𝜙_‡

Tunable-coupler: Flux-noise sensitivity

Qubit coherence

limited by flux noise on tunable coupler: S ω = A^./ωwith A = (3.9 ± 0.7) 10^6w ϕ_‡

Tunable coupler coherence (measured)

𝑇_- (µs) 𝑇_.^∗ (𝜇𝑠)

𝜙 = 0 13.5 15

𝜙 = −0.108 𝜙_‡ 12 0.5

Spectroscopy

Device

Tunable-coupler: 2-qubit gates

State tomography

Qubit-qubit exchange rate: 𝛺

∼

¸,¹

→ Larger rate for iSWAP (< 20 MHz) than bSWAP (< 5 MHz)

»00⟩ + |11⟩ via bSWAP (π/2 in 160 ns)

Fidelity: 97.4%

»10⟩ + |01⟩ via iSWAP (π/2 in 80 ns) Fidelity: 97.8%

§ Full Hamiltonian of qubit and coupler:

𝐻^¼½¾¿À = 𝜔 𝑎_-^N𝑎_-+^-_. + ^-_. 𝛼𝑎_-^N𝑎_- 𝑎_-^N𝑎_-− 1 and 𝐻^{Á½ÃÄ¿ÅÆ} = 𝑔_-. 𝑎_-^N+ 𝑎_- 𝑎_.^N+ 𝑎_.

Tunable coupler: Numerical simulation of transitions

000

100 010

001

4.40GHz

0GHz

4.97GHz 6.02GHz

Δ δ/𝟑

coupler

difference gate

coupler

coupler

coupler

numerical simulations based on Qutip

§ iSWAP gate (0.57 GHz): close to higher-order qubit-coupler transitions causing leakage

§ bWAP gate might be preferable

Outlook

§ Implement the adiabatic protocol for the hydrogen molecule in experiment