© simon hönl quantum chemistry calculations on a ... · ibm q [email protected] –symposium h....
TRANSCRIPT
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)
IBM Q [email protected] – Symposium H. Rauch, May 2019
• 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
IBM Q [email protected] – Symposium H. Rauch, May 2019
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
IBM Q [email protected] – Symposium H. Rauch, May 2019
Superconducting Qubit
6
tunable non-linear inductance:
𝐿 𝜙 =𝜙0/2𝜋
𝐼𝑐 cos𝜋𝜙𝑒𝑥𝑡𝜙0
Capacitance C
ෝ=
superconductingelectrode
𝐿(𝜙) C
IBM Q [email protected] – Symposium H. Rauch, May 2019
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:
IBM Q [email protected] – Symposium H. Rauch, May 2019
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:
IBM Q [email protected] – Symposium H. Rauch, May 2019
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
IBM Q [email protected] – Symposium H. Rauch, May 2019
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
IBM Q [email protected] – Symposium H. Rauch, May 2019
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
IBM Q [email protected] – Symposium H. Rauch, May 2019
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
© 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
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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!
© 2018 International Business Machines Corporation
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/
© 2018 International Business Machines Corporation
𝐻2: 2 qubits LiH: 4 qubits 𝐵𝑒𝐻2: 6 qubits
Quantum Computing for Quantum Chemistry
[A. Kandala, et al. Nature 549 (2017)]
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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 | ↓⟩
© 2018 International Business Machines Corporation
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)
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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
© 2018 International Business Machines Corporation
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)]
© 2018 International Business Machines Corporation
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⟩
© 2018 International Business Machines Corporation
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)]
IBM Q [email protected] – Symposium H. Rauch, May 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):