© Simon Hönl Quantum chemistry calculations on a ... · IBM Q [email protected] –Symposium H....

Stefan FilippIBM Research – Zurich

Switzerland

Quantum chemistry calculations on a superconducting

qubit quantum processor

Quantum Forever Symposium, Atominstitut, Wien – May 22, 2019

© Simon Hönl

IBM Q [email protected] – Symposium H. Rauch, May 2019

Acknowledgments

IBM Research Zurich

Experiment:Daniel EggerMarc GanzhornGian SalisMax WerninghausAndreas FuhrerPeter Müller

Theory:Panagiotis BarkoutsosPauline OllitraultNikolaj MollIvano Tavernelli

2

IBM Research Yorktown

D. McKay, V. Adiga, A. Mezzacapo, J. Chow, J. Gambetta

University CollaboratorsM. Roth, D. DiVincenzo (RWTH Aachen)

S. Schmidt (ETH Zurich)


• Many computational paths from initial state to each final state

• Each path accumulates a complex phase, e.g. 1,−1, 𝑖, 𝑒𝑖𝜋/4, …• Output probability: concentrated at the final states where (almost) all paths arrive with

(approximately) the same phase.

Quantum Computing: Extra power from interference

IBM Q

Easy Problems

13 x 7 = ?

937 x 947 = ?

Hard Problems for

Classical Computing (NP)

Possible with

Quantum Computing

Material,

Chemistry

Machine

Learning

Optimization

Many problems in business and science are too complex for classical computing systems

“hard” / intractable problems:(exponentially increasing resources with problem size)

• Algebraic algorithms (e.g. factoring, systems of equations) for machine learning, cryptography,…

• Combinatorial optimization (traveling salesman, optimizing business processes, risk analysis,…)

• Simulating quantum mechanics (chemistry, material science,…)

91 = ? x ?

887339 = ? X ?

Quantum Computing as a path to solve intractable problems


Interferometry with massive particles

[H. Rauch, W. Treimer and U. Bonse, PLA 47 369 (1974).]

flux quanta [𝜙0]

crit

ical

curr

ent

[𝐼𝑐]

Double-slit InterferenceNeutron Interferometer SQUID (SuperconductingQuantum Interference Device)

slit

po

siti

on

intensity

Josephson junction

𝐼𝑐 = 𝐼0 cos𝜋𝜙

𝜙0


Superconducting Qubit

6

tunable non-linear inductance:

𝐿 𝜙 =𝜙0/2𝜋

𝐼𝑐 cos𝜋𝜙𝑒𝑥𝑡𝜙0

Capacitance C

ෝ=

superconductingelectrode

𝐿(𝜙) C


harmonic LC oscillator Coplanar waveguide resonator = photon

Quantum Electronic Circuits: Harmonic Oscillator

classical: quantum:

𝐻 =𝜙2

2𝐿+𝑞2

2𝐶

𝜔 =1

𝐿𝐶~ 6 𝐺𝐻𝑧

𝐻 =𝜙2

2𝐿+ො𝑞2

2𝐶= ℏ𝜔 ො𝑎† ො𝑎 +

1

2

𝜙, ො𝑞 = 𝑖ℏ |0⟩

|1⟩

|2⟩

|3⟩

|4⟩

…

basic circuit elements:


Josephson junction circuit = qubit

An-harmonic quantum oscillator (=qubit)

Josephson junction:

𝐻 =ො𝑞2

2𝐶+

𝐼𝑐2𝜋𝜙0

cos መ𝛿

= ℏ 𝜔 + 𝛼 ො𝑎† ො𝑎 ො𝑎† ො𝑎 +1

2 qu

bit

𝐿𝐽 𝛿 =

𝜙0/2𝜋

𝐼𝑐 cos 𝛿

𝝎𝒒

An-harmonic LC oscillatorbasic circuit elements:


Frequency-tunable superconducting quantum circuit (=qubit)

𝐿𝐽 𝜙𝑒𝑥𝑡𝐵𝑒𝑥𝑡

𝐶

Charging Energy: EC =(2𝑒)2

2𝐶

Josephson Energy: 𝐸𝐽 𝜙𝑒𝑥𝑡 ∝ 1/𝐿(𝜙𝑒𝑥𝑡)

(Transmon) transition frequency:

𝜔𝑞 = 8𝐸𝐽𝐸𝐶 − 𝐸𝑐

Usage:• Frequency tunable qubit

(sensitive to B-field noise)• Tunable coupler

basic circuit elements: Tunable, an-harmonic LC oscillator


Superconducting Qubit Processor

(Tunable) Microwave resonator as:▪ read-out of qubit states▪ quantum bus▪ noise filter

Superconducting qubit:▪ fixed-frequency transmon-type qubits▪ T1, T2 ~ 70 µs lifetime, 10-500ns gate time


5 Qubits (2016)16 Qubits (2017)

IBM Q experience (publicly accessible)

IBM Q commercial

IBM qubit processor architecturesIBM Research – Zurich (experimental)

Tunable coupler architecture:• High-fidelity 2-qubit gates• Mitigation of frequency crowding• Larger gate-set


research.ibm.com/ibm-qx

Since launch

• > 120,000 users

• > 150,000 Qiskit downloads

• > 9,000,000 experiments

• > 150 research papers

• used by 1,500+ colleges and universities, 300 high schools, 300 private institutions

IBM QX Features

• Tutorial

• Simulation

• Graphical programming

• QASM language

• API & SDK

• Active user community

Experience quantum computing here:

Public quantum computer (currently up to 14 qubits) and developer ecosystem

IBM Quantum Experience

http://research.ibm.com/ibm-qx/

© 2018 International Business Machines Corporation

How much memory is needed to store a quantum state?How much time does it take to calculate dynamics of a quantum system?

# qubits quantum state coefficients # bytes timescale

1 𝑎 0 + 𝑏|1⟩ 21 = 2 16 Bytes

2 𝑎 00 + 𝑏 01 + 𝑐 10 + 𝑑|11⟩ 22 = 4 32 Bytes Nanoseconds

8 28 = 256 2kB Microseconds on watch

16 … 216 = 65′536 512 kB Milliseconds on smartphone

32 … ~4 billion 32 GB Seconds on laptop

64 …~ information

in internet128 EB

(134 million GB)Years on supercomputer

256 …~ # of atoms in

universe… never

The Quantum Advantage – Storing quantum states


reaction ratesmolecular structure

Interacting fermionic problems: is at the core of most challenges in computational physics and high-performance computing(sign problem: Monte-Carlo simulations of fermions are NP-hard [Troyer &Wiese, PRL 170201 (2015)])

Quantum computer: map fermions (electrons) to qubits to compute

Application: Quantum chemistry

𝐻𝑒 = −

𝑖=1

𝑁1

2𝛻𝑖2 −

𝑖=1

𝑁

𝐴=1

𝑀𝑍𝐴𝑟𝑖𝐴

+

𝑖=1,𝑗,𝑖>1

1

𝑟𝑖𝑗


Quantum chemistry – Classical approximationsClassically, several approximations have been derived to break the exponential scaling

Hartree Fock(HF)

MP2 MP3 CCSD

CCSD(T)

DFT

N3

N4

N5

N6

N7

Ab-initio methods

Increasing accuracyMP: Moller-Plesset

CC: Coupled Cluster

DFT: Density FunctionalTheory

N: Number of electrons(basis functions)Fi

rst-

pri

nci

ple

CCSDT

N2

N3

Variational method

Task: find the ground state of a Hamiltonian 𝐻𝑝𝑟𝑜𝑏

Method: minimize energy function E = 𝜓 𝜃 𝐻𝑝𝑟𝑜𝑏 𝜓 𝜃 → 𝑚𝑖𝑛.

Hamiltonian problem

Map the problem to qubits

(e.g. hydrogen molecule 𝐻𝑞 = 𝛼1 𝜎𝑧(1)

+ 𝛼2 𝜎𝑧2+ 𝛽𝜎𝑧

1𝜎𝑧

2+ 𝛾𝜎𝑥

1𝜎𝑥

2)

Prepare trial state 𝜓 𝜃 ;measure qubits

Solution 𝐸𝑚𝑖𝑛(𝜃𝑚𝑖𝑛)

Hybrid Quantum-classical computer

[Barrett 2013; Farhi, 2014; Peruzzo 2014; O’Malley 2015; McClean 2016; Eichler 2016; Kandala 2017]

evaluate 𝑬 𝜽 ;classical optimizer to choose new 𝜃 until 𝑬𝒎𝒊𝒏 is found


Preparation of trial state

• Parameterized single qubit gates

(𝜃… rotation angles)

• Two-qubit entangling gates (CNOTs)

Target state: 𝜓 𝜃 = 𝑈𝑒𝑛𝑡𝑈0 𝜃 … [𝑈𝑒𝑛𝑡𝑈

𝑑 𝜃 ]|0𝑁⟩

Measurement of energy (= cost function): 𝐸 𝜃 = 𝜓 𝜃 𝐻𝑚𝑜𝑙 𝜓 𝜃

e.g. for hydrogen molecule 𝐻𝐻2 = 𝛼1 𝜎𝑧1

+ 𝛼2 𝜎𝑧2

+ 𝛽 𝜎𝑧1𝜎𝑧

2+ 𝛾 𝜎𝑥

1𝜎𝑥

2

Option: Heuristic method using available gates

NOTE: Target state is not specific to (chemistry) problem!


Energy as sum of Pauli operators 𝑃𝛼

𝐸 Ԧ𝜃 = ⟨𝜓 Ԧ𝜃 𝐻 𝜓 Ԧ𝜃 ⟩ = σ𝑎 ℎ𝛼 𝑃𝛼

with Pauli-strings 𝑃𝛼 = 𝜎𝑥,𝑦,𝑧 ⊗𝜎𝑥,𝑦,𝑧 ⊗⋯𝜎𝑥,𝑦,𝑧

Protocol:

1. measure individual Pauli strings for pulse

parameters 𝜃𝑖 = {𝜃𝑖1, 𝜃𝑖

2, … , 𝜃𝑖𝑝}

2. Optimize Ԧ𝜃𝑖 to find minimum energy 𝐸𝑚𝑖𝑛 ( Ԧ𝜃𝑚𝑖𝑛)

Energy measurement and optimization protocol

Image source: John Hopkins Applied Physics Laboratory, http://www.jhuapl.edu/


𝐻2: 2 qubits LiH: 4 qubits 𝐵𝑒𝐻2: 6 qubits

Quantum Computing for Quantum Chemistry

[A. Kandala, et al. Nature 549 (2017)]


Error Mitigation – Richardson Extrapolation

[A. Kandala, et al. Nature 567 (2019)]

Influence of noise scales with the length of the gate→ perform experiment with different gate lengths→ extrapolate to gate length zero |0⟩

|0⟩

𝑈1𝑈𝑒𝑛𝑡

𝑈1

𝑈1

𝑐1𝜆

𝑈1𝑈𝑒𝑛𝑡

𝑈1

𝑈1

𝑐2𝜆


Problem-specific trial states for H2

• Molecular wavefunction: superposition of states |𝑛1𝑛2𝑛3𝑛4⟩ with occupation 𝑛𝑖 ∈ {0,1} of mode 𝜒𝑖[Whitfield et al. (2010)]

• Relevant subspace for H2: N=2, states 1100 , 1010 , 1001 , 0110 , 0101 , |0011⟩

• Groundstate: superposition of basis states

ΨG = 𝜃1 1100 + 𝜃2 0110 + 𝜃3 1001 …

• Exchange-type gates 𝑈𝑒𝑥 preserve # of excitations:

ΨG = 𝑈𝑒𝑥𝑛 𝑈𝑒𝑥

𝑛−1…𝑈𝑒𝑥1 |1100⟩

• for H2: zero-magnetic moment subspace is mapped to 2 qubits:

⟩𝛹𝑒𝑥 𝜃, 𝜙 = 𝑈𝑒𝑥 𝑋𝑝 ⟩00 = 𝑐𝑜𝑠 𝜃 01 − 𝑖 𝑠𝑖𝑛 𝜃 𝑒−𝑖𝜙|10⟩

+ +

𝜙𝑔, ↑

, + +

𝜙𝑔, ↓

+ +

𝜙𝑢, ↑

+ +

𝜙𝑢, ↓

, ,

𝜒1 = 𝜙𝑔 | ↑⟩ 𝜒2 = 𝜙𝑔 | ↓⟩ 𝜒3 = 𝜙𝑢 | ↑⟩ 𝜒4 = 𝜙𝑢 | ↓⟩


Reduction of gate countGoal: Finish algorithm within the coherence time of the superconducting quantum hardware

Problem-specific trial states (exchange-type gates):# of qubit excitations conserved ↔ particle conservation e.g. by decomposition into natural gates [Barkoutsos et al., Phys. Rev. A (2018)]

Direct implementation in hardware:Use exchange-type gates (e.g. iSWAP)[Ganzhorn et al., PR Applied (2019)]

Variational form using CNOT entanglers:not specific to problem, overhead in number of gates,target state may be hard to reach

Circuit depth required to reach chemical accuracy (6.5 mHa) (without gate errors)


Hardware implementation of exchange-type gates

Parametric frequency modulation of tunable coupler:[Bertet et al., PRB (2006); Niskanen et al., Science (2007); Tian et al., NJP (2008); Kapit, PRA (2013); Roushan, Nature Physics (2017); Didier et al., PRA 97 (2018); etc.]

Apply time-dependent magnetic flux Φ 𝑡 at qubit difference frequency 𝜔Φ = 𝜔1 −𝜔2

Hamiltonian in rotating frame:

𝐻𝑒𝑓𝑓 = −Ω𝑒𝑓𝑓4

[cos 𝜑 𝜎𝑥𝜎𝑥 + 𝜎𝑦𝜎𝑦 + sin𝜑 (𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑥)]

Exchange interaction |10⟩ ↔ 01 with tunable rate 𝛺𝑒𝑓𝑓 & phase 𝜑[McKay et al., Phys. Rev. Applied (2016), Roth et al., Phys. Rev. A (2017)]

𝜔Δ

| ⟩10

| ⟩11

| ⟩01

| ⟩00

𝜔𝑐 𝑡 = 𝜔0 |cos 𝜋Φ 𝑡 /Φ0 |

|1⟩𝑡𝑐


Hardware implementation of exchange-type gates

qubit 𝜈0→1𝑚𝑎𝑥 [GHz] 𝑇1 [us] 𝑇2 [us] 𝑇2

∗ [us]

Q1 4.959 78 97 86

Q2 6.032 23 23 13

TC 7.333 7.5 0.08 0.02

Apply time-dependent magnetic flux Φ 𝑡 at qubit difference frequency 𝜔Φ = 𝜔1 −𝜔2

Hamiltonian in rotating frame:

𝐻𝑒𝑓𝑓 = −Ω𝑒𝑓𝑓4

[cos 𝜑 𝜎𝑥𝜎𝑥 + 𝜎𝑦𝜎𝑦 + sin𝜑 (𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑥)]

Exchange interaction |10⟩ ↔ 01 with tunable rate 𝛺𝑒𝑓𝑓 & phase 𝜑[McKay et al., Phys. Rev. Applied (2016), Roth et al., Phys. Rev. A (2017)]

𝜔Δ

| ⟩10

| ⟩11

| ⟩01

| ⟩00

|1⟩𝑡𝑐


Exchange-type interaction

𝑈𝒆𝒙 = 𝒆−𝒊ℏ𝑯𝒆𝒇𝒇 𝜽,𝝓 =

1 0 0 00 cos 𝜃 −𝑖 𝑒−𝑖𝜙sin 𝜃 00 −𝑖 ei𝜙sin 𝜃 cos 𝜃 00 0 0 1

Unitary matrix:

→ Creation of arbitrary 10 - 01 rotation 𝑈𝑒𝑥(𝜃, 𝜑) with currently ~ 95% fidelity (verified also via RB)

𝜙: phase of the gate

(controlled by phase of pulse)

𝜃: population transfer from 10 to 01

(controlled by length of pulse)

𝜔Δ

| ⟩10

| ⟩11

| ⟩01

| ⟩00

QPT fidelity QPT fidelity

|1⟩𝑡𝑐


Energy eigenstates of the 𝐻2 molecule

Ground state GVQE trial states

𝜓 𝜃, 𝜑 = 𝑎 𝜃, 𝜑 01 + 𝑏 𝜃, 𝜑 10= 𝑈𝑒𝑥 𝜃, 𝜑 |01⟩

Accuracy limited by coherence time of tunable coupler (𝑇2

∗~25 𝑛𝑠) Chemical accuracy

meas|0⟩

meas|𝟎⟩

𝑋𝜋

𝑈𝑒𝑥 𝜃, 𝜙

𝑋𝑋𝑋

𝑋𝑋𝑋

measure XX, ZZ,…

SPSA optim.

2-qubit Hamiltonian:

𝐻𝐻2 = 𝛼0𝐼𝐼 + 𝛼1𝑍𝐼 + 𝛼2𝐼𝑍 + 𝛼3𝑍𝑍 + 𝛼4𝑋𝑋

(𝛼𝑖 encode bond length)

[Ganzhorn et al., PR Applied (2019)]


Excited states – Equation-of-Motion (EOM) Method

Goal: Calculate excitation energies

En0 = En − E0 =0 𝑂𝑛,[𝐻,𝑂𝑛

†] 0

0 𝑂𝑛,𝑂𝑛† 0

Method:

• Express excitation EOM operators M ∈ 𝑂𝑛, [𝐻, 𝑂𝑛†] in terms of Pauli-strings

(by solving a set of linear equations on classical computer):

M = σ𝑎 𝑔𝛼 𝑃𝛼 with 𝑃𝛼 = 𝜎𝑥,𝑦,𝑧 ⊗𝜎𝑥,𝑦,𝑧 ⊗⋯𝜎𝑥,𝑦,𝑧

• Measure Pauli-strings in A for given ground state |0⟩ (from VQE)

𝑂𝑛†

[classical: J. F. Stanton & R. J. Bartlett, J. Chem. Phys. (1993); quantum: Ollitrault et al., in preparation (2019)]

|0⟩

|𝑛⟩


Energy eigenstates of the 𝐻2 molecule

Excited states E1,E2,E3 of 𝐻2:

Similar accuracy as ground state energy

Can be scaled to larger systems (EOM scales with 𝒪(𝑁8))

[Ollitrault et al., in preparation (2019)]

Stable alternative to Quantum Subspace Expansion[Colless et al. PRX 8 (2018)]

Chemical accuracy

[Ganzhorn et al., PR Applied (2019)]


Outlook on quantum chemistry

Improved coherence time of the TC:Chemical accuracy for 𝐻2 reached at 𝑇2

∗~500 ns

Compute the full energy spectra of larger molecules:

Multi-qubit tunable coupler architecture with 𝑁𝑞𝑢𝑏𝑖𝑡 ≥ 4

This work

TC

qubit

More efficient VQE algorithm (higher rep rates,efficient optimizers optimal control):

© Simon Hönl Quantum chemistry calculations on a ... · IBM Q [email protected] –Symposium H....

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