Towards an experimentally viable variational quantum eigensolver with superconducting qubits

Ryan BabbushGoogle Inc.

January, 2016

Goal: build quantum computer that does something useful, ASAP

The dream of error correction is worth pursuing

The road to fault tolerance is long and hard;Many “universal but analog” devices lie ahead

Can such platforms perform any useful computation?The question is inevitable for devices of certain scale

Claim: variational algorithms applied to fermions may solve classically intractable problems on such devices

Meet the hardware: Xmons and Gmons

Demonstrated bit-flip error correction on Xmon

To entangle, qubit frequencies adjusted to resonance

Gmon like Xmon; has square shape and adjustable coupler

Can leave qubits at single frequency, avoid crowding

Gmon is not replacement for Xmon; we plan to scale both

Xmon ReferencesR. Barends et al. Nature 508, 500-503 (2014).J. Kelly et al. Nature 519, 66-69 (2015).

Gmon ReferencesY. Chen et al. PRL 113, 220502 (2014). P. Roushan et al. Nature 515, 241–244.

Gmons have different Hamiltonian, more control

Inductive coupler gives time-dependent Hamiltonian,

“flip-flop” gate generated by XX + YY interaction,

Quantum supremacy with shallow circuits?

Working towards planar array to implement surface code

Cost to simulate 9 by 9 array with gate depth 9 is approximately O(281) = O(1024)

To avoid decoherence, we must limit to shallow circuits

Cost of classical simulation depends exponentially on treewidth of space time circuit graph

I. Markov and Y. Shi. SIAM Journal on Computing 38:3 (2008)

What to do with these “powerful” shallow circuits?

get new parameters

What is the value of being able to prepare complex quantum states?Such states can provide classically inaccessible descriptions of quantum systems

Ground states of small fermionic models often lack adequate classical descriptionse.g. quantum chemistry and Fermi-Hubbard model

But how do we make the right states?Use circuit to form variational ansatz!

Training loop robust to systematic errors

Quick history of the variational quantum eigensolver (VQE)

1. First paper on VQE and implementation with quantum optics (April 2013):A. Peruzzo, J. McClean, P. Shadbolt, M. Yung, X. Zhou, P. Love, A. Aspuru-Guzik, J. O'BrienNature Communications, 5:4213, (2014)

2. Theoretical implementation with ion trap (July 2013)M. Yung, J. Casanova, A. Mezzacapo, J. McClean, L. Lamata, A. Aspuru-Guzik, E. SolanoScientific Reports, 4:3589 (2014)

3. Analysis of measurements needed for chemistry (July 2014)J. McClean, R. Babbush, P. Love, A. Aspuru-GuzikJournal of Physical Chemistry Letters, 5 (24): 4368–4380 (2014)

4. First implementation with ion trap (June 2015)Y. Shen, X. Zhang, S. Zhang, J. Zhang, M. Yung, K. KimarXiv preprint: 1506.00443

5. Application to Fermi-Hubbard and numerics (July 2015):D. Wecker, M. B. Hastings, M. TroyerPhysical Review A, 92:042303 (2015)

6. First implementation with superconducting qubits (August 2015):C. Eichler, J. Mlynek, J. Butscher, P. Kurpiers, K. Hammerer, T. Osborne, A. WallraffPhysical Review X, 5:041044 (2015)

7. Theory generalization and error robustness (September 2015):J. McClean, J. Romero, R. Babbush, A. Aspuru-GuzikNew Journal of Physics 18 (2): 023023 (2016)

8. First scalable quantum chemistry simulation (December 2015):P. O'Malley, R. Babbush, I. Kivlichan, J. Romero, J. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, P. Coveney, P. Love, H. Neven, A. Aspuru-Guzik, J. MartinisarXiv preprint: 1512.06860

The molecular electronic structure problem

Goal is to solve for the energy of a molecule:

Energy surfaces allow us to understand reactions

For chemical rates we need chemical accuracy

Clamp nuclei under Born-Oppenheimer approximation, write in second quantization:

Mean-field state is within 1% of correct energy

Variational quantum eigensolver (VQE) applied to H2

1. Cleverly parameterize short quantum circuit with a polynomial number of variables

2. Apply circuit to guess state and measure energy

3. Use classical optimizer to suggest new parameters

Scalable quantum simulation of molecular energies

Used Xmon qubits to compute energy surface of molecular hydrogenStarted in Hartree-Fock state, used unitary coupled cluster, got chemical accuracy

P. O’Malley, R. Babbush et al. arXiv preprint: 1512.06860

Predicted dissociation energy without exponentially costly compilation for first timeSubstantial robustness to systematic errors seen

Scalable quantum simulation of molecular energies

P. O’Malley, R. Babbush et al. arXiv preprint: 1512.06860

VQE with a “sublogical hardware ansatz”

Individual gates usually calibrated in VQE’esk fashion

Parameterize at level of hardware for power and robustness

Need to think carefully about control parameterizations

New paradigm: more parameters, better solutions, hard optimization?

Currently preparing VQE experiment on Fermi-Hubbard model

Numerical studies to investigate performance of VQE in open system

Thanks to many collaborators, in particular:

Xmon Experiment Gmon Experiment Theory Support

Peter O’Malley (UCSB) Jimmy Chen (UCSB) Alan Aspuru-Guzik (Harvard)

Rami Barends (Google) Pedram Roushan (Google) Jarrod McClean (LBNL)

Scalable Quantum Simulation of Molecular Energies

Able to get chemical accuracyusing single gate

Representation in Occupation Number BasisWe now write the real space Hamiltonian in a second quantized spin-orbital basis

Unitary Coupled Cluster Ansatz

Hilbert space is vast - don’t get lost

Unitary coupled cluster = classically intractable, extremely powerful

• Use perturbation theory to truncate terms

• Chemical accuracy with single parameter for hydrogen

Why should we care about quantum chemistry?

Humans: Haber-Bosch process, N

2 + 3 H

2 → 2 NH

3 500°C, 20 MPa

Consumes 2% of world energy

2013 Waco, Texasfertilizer plant explosion

Nature: Nitrogenaseaka “MNIST for QNNs”

N2 + 3 H

2 → 2 NH

3 25°C, 0.1MPa

Fe2S

2 center cannot be

simulated (168 qubits)Fertilizer sales in 2006:

$72,000,000,000

green sulphur bacteria>90% efficient