Quantum Chemistry using SC Qubits 09 04 2018

Quantum Chemistry using SC Qubits 09. 04. 2018 Giulia Mazzola, Simon Mathis, Thilo Schmid

Quantum Chemistry Quantum mechanics → chemical properties i. e. solve 09. 04. 2018 Thilo Schmid, ETH Zurich

Born-Oppenheimer Approximation constant depending on R small 09. 04. 2018 Thilo Schmid, ETH Zurich

Energy Surface Example - Water molecule q 1 q 2 https: //en. wikipedia. org/wiki/Potential_energy_surface 09. 04. 2018 Thilo Schmid, ETH Zurich

Energy Surface E. G. Lewars, Computational Chemistry (Springer, 2011) → compute reaction rates (Eyring equation) need chemical accuracy! 09. 04. 2018 Thilo Schmid, ETH Zurich

Problems of Classical Computers Computational complexity scales exponentially with the problem Example – n two-level systems 09. 04. 2018 Thilo Schmid, ETH Zurich

(…) and I'm not happy with all the analyses that go with just the classical theory, because nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. - Richard Feynman, Simulating Physics with Computers, 1981 published in Int. J. Theor. Phys. 21 (1982) 09. 04. 2018 Thilo Schmid, ETH Zurich

Digital Quantum Simulation Idea: (1) Map problem to qubit system (2) Solve qubit problem on the quantum computer (3) Patch together solution on classical computer Limited by technology 09. 04. 2018 Error correction possible Thilo Schmid, ETH Zurich

Second Quantization Classical computer Quantum computer Overlap integrals using basis sets 09. 04. 2018 Thilo Schmid, ETH Zurich

Jordan Wigner • Map the fermions to • Imitate the anti-commutation relations but 09. 04. 2018 Thilo Schmid, ETH Zurich

Jordan Wigner yields 09. 04. 2018 Thilo Schmid, ETH Zurich

Jordan Wigner yields non-local 09. 04. 2018 → use Bravyi-Kitaev instead Thilo Schmid, ETH Zurich

Reduces, because we can compute Z 1 and Z 3 directly from Hartree-Fock state 09. 04. 2018 Thilo Schmid, ETH Zurich

Overview: Does the same job as Jordan Wigner but for O(log n) instead of O(n) qubits 09. 04. 2018 Thilo Schmid, ETH Zurich from P. J. J. O’Malley et al. , Phys. Rev. X 6 (2016)

Workflow Quantum Algorithm? P. J. J. O’Malley et al. , Phys. Rev. X 6, 031007 (2016)

Phase estimation algorithm (PEA) Main steps: 1) Prepare ground state 2) Apply controlled time evolution 3) Measure phase of ancilla Trotterization:

Variational quantum eigensolver (VQE) ground state minimizes the energy Key point: Find ground state by minimizing w. r. t.

Variational quantum eigensolver

Overview

Parametrization of the ground state wavefunction • Hamiltonian • Variational Principle 1 st Idea: , Hartree-Fock Approximation Then 2 nd Idea (VQE): is small, but crucial for chemistry! ,

Parametrization of the ground state wavefunction , quantum circuit What is the explicit form of ? Unitary Coupled Cluster Theory: quantum gates

How does the experimental implementation look like?

Xmon Transmon Qubits Implementation P. J. J. O’Malley et al. , Phys. Rev. X 6, 031007 (2016)

Xmon Transmon Qubits Implementation microwave pulse sequences (single-qubit rotations) P. J. J. O’Malley et al. , Phys. Rev. X 6, 031007 (2016) dc pulses (two-qubit entangling gates)

Results H 2 energy curve minimum energy bond length (R = 0. 72 Å) P. J. J. O’Malley et al. , Phys. Rev. X 6, 031007 (2016) dissociation energy

Results H 2 energy curve • the dissociation energy determines the chemical dissociation rate • The efficient implementation predicts the correct dissociation energy to within chemical accuracy (small errors) • The VQE is robust to systematic device errors • minimum energy bond length (R = 0. 72 Å) dissociation energy P. J. J. O’Malley et al. , Phys. Rev. X 6, 031007 (2016) first scalable quantum simulation of molecular energies performed on quantum hardware

More experiments (IBM) A. Kandala et al. , Nature 549, 242– 246 (2017)

More experiments (IBM) A. Kandala et al. , Nature 549, 242– 246 (2017)

Thank you! 09. 04. 2018 Giulia Mazzola, Thilo Schmid, Simon Mathis, ETH Zurich 31

Backup Slides

Bravyi – Kitaev Transform • from 2 nd quantized Hamiltoian Coeff. computed from orbital overlap (after choice of basis) • need to satisfy • iterative definition (BK-Transform)

Bravyi – Kitaev Example

Electronic structure problem • Molecular Hamiltonian (in Hartee) • Hartee:

Electronic structure problem – 2 nd quantization

Unitary coupled cluster ansatz (UCC) • Example: • BK transformed H 2 Hamiltonian • Unitary coupled cluster theory ansatz:

Variational quantum eigensolver (VQE) ground state minimizes the energy Example: H 2 Hamiltonian (qubit-form)