Algorithmic simulation of far-from- equilibrium dynamics using quantum computer Walter V. Pogosov 1,2,3 1 Dukhov Research Institute of Automatics (Rosatom), 2 Institute for Theoretical and Applied Electrodynamics RAS, 3 Moscow Institute for Physics and Technology In collaboration with A. A. Zhukov, S. V. Remizov, and Yu. E. Lozovik

Outline Motivation Simulation of central spin model with 5- qubit quantum computer Simulation of transverse-field Ising model with 16-qubit quantum computer Summary

Motivation Dramatic progress in the construction of quantum computers and simulators using different physical realizations (superconducting Josephson circuits, trapped ions, neutral atoms, etc.) Quantum computers and simulators are believed to be extremely useful in simulation of many-body systems: novel materials, quantum chemistry, drugs. Arguments: first-principle simulation of quantum systems is difficult due to the exponential explosion of Hilbert space size (2^N states for spin systems).

Two types of quantum computing devices 1.Algorithmic quantum computer A set of discrete operations. The same processor can be used to realize many different algorithms, including quantum modeling of different systems. Different physical platforms. Superconducting circuits and trapped ions are leading for the moment. Thousands of operations already can be made within the decoherence time 20-qubit IBM device 72-qubit Google device Nontrivial physics begins with tens of qubits (2^50 states is too many for most powerful modern supercomputers) Are we close to some practical impact?

2. Analog simulator One-to-one correspondence between the modeled system and simulator (spins = qubits). Suitable for a single problem or a narrow class of problems. Ising model – ground state. Optimization (reduction to Ising).

- Bottleneck: fermionic Hamiltonians molecules, strongly correlated materials (Hubbard model etc); most interesting due to the sign problem in Monte-Carlo methods.

Jordan-Wigner transformation Let us numerate all sites (physical qubits) and define collective operators Correct commutation relations! The price is an extreme complexity of the Hamiltonian in spin variables (1D is an exception) Bravyi-Kitaev transformation – the situation is better, but…

Spin Hamiltonians -If we want to model a spin system (some magnetic materials) Spin Hamiltonians, for example: Ising model in a transverse field - One-to-one mapping between spins and qubits. - Spin ladder (IBMqx5 chip).

Unitary evolution. General scheme. I. Representation of Hamiltonian through spin variables. Correspondence between the degrees of freedom of modeled system and qubits of the chip. II. Preparation of the initial state in real quantum device III. Simulation of unitary (free) evolution in real quantum device using Trotterization (discretization) of evolution operator IV. Possible determination of eigenenergies from the dynamics using phase estimation algorithms Bottleneck: many gates, very long algorithms

Unitary evolution. General scheme. I. Representation of Hamiltonian through spin variables. Correspondence between the degrees of freedom of modeled system and qubits of the chip. II. Preparation of the initial state in real quantum device III. Simulation of unitary (free) evolution in real quantum device using evolution operator and its Trotterization via discrete operations IV. Possible determination of eigenenergies from the dynamics using phase estimation algorithms Bottleneck: many gates, very long algorithms

Far-from-equilibrium dynamics Nonequilibrium quantum relaxation in closed many-body systems. Quenches in trapped cold-atom gases (instantaneous change of parameters). Central issues: 1.Whether the system relaxes to a stationary state (“thermalization”)? What are its characteristics? 2.Dynamical evolution of order, correlations, entanglement. - Depends on the integrability of the model - Depends on the initial state - Finite particle number effects …

Far-from-equilibrium dynamics Aims of this work: Main message: Algorithmic quantum simulation is attractive. The same chip can be used for simulation of different spin models and different initial conditions (much more flexible than quenches). No need for the phase estimation algorithm – only dynamics. Even semi-quantitative accuracy might be valuable… The closest real “application” of quantum computers in quantum modeling? Proof-of-principles experiments, which unveil capabilities of modern quantum computers.

Basic tools of quantum computation Single qubit gates Examples Fidelities are already very high – 0.999…

Two-qubit gates An example of the two-qubit gate Readouts Bottleneck: errors in multiqubit chips are relatively large, > 1 %. Progress is very slow if any. Scaling? Errors are also relatively large, > 1 %. Progress is slow. Errors is another crucial characteristics, which is complementary to T 1 and T 2

Direct mapping between degrees of freedom of a modeled system and degrees of freedom of the physical qubits of the chip II. Central spin model and 5-qubit device - Interaction between particles is implemented using CNOTs -CNOTs are also utilized to create initial states. -This geometry is potentially useful in the framework of DMFT quantum-classical calculations (impurity problem solver)

Full resonance (in the rotating frame) Another physical realization: Dicke Hamiltonian (quantum optics) Resonance + rotating wave approximation Sector with single excitation – exact mapping to XX central spin model

Unitary evolution. General scheme. I. Representation of Hamiltonian through spin variables. Correspondence between the degrees of freedom of a modeled system and qubits of the chip. II. Preparation of the initial state in real quantum device III. Simulation of unitary (free) evolution in real quantum device using evolution operator and its Trotterization via discrete operations

Initial state of the system – entangled “bath” tunable phase parameter. Dynamics of the central spin can be suppressed due to the negative quantum interference of contributions from two qubits Preparation of the initial state in real quantum device - Cancellation of two contribution coming from two different spins. - No central spin dynamics. “Dark” state from quantum optics. - Еxcitation blockade in the bath due to the quantum interference.

CNOTs are not very good (errors) Quantum circuitEncoding Encoding initial condition to the chip

Proof that this circuit gives desirable result

Unitary evolution. General scheme. I. Representation of Hamiltonian through spin variables. Correspondence between the degrees of freedom of modeled system and qubits of the chip. II. Preparation of the initial state in real quantum device III. Simulation of unitary (free) evolution in real quantum device using evolution operator and its Trotterization via discrete operations

Free evolution (through evolution operator) Modeling dynamics This representation is needed for quantum computer and not for us! Trotter-Suzuki decomposition exact in the limit The larger number of Trotter steps, the smaller (mathematical) Trotterization error

One-step Trotter decomposition Example of the circuit easy to check directly

Main building block for modeling interaction

Full quantum circuit

Dark and bright states known from quantum optics Entanglement in the bath and quantum interference effects block excitation transfer to the center A method to find Hamiltonian eigenstate Two-particle entangled state: Population of the central particle experiment (8000 runs) theory Attention! Theory is not exact. Approximation of the same level – one-step Trotter decomposition

Heuristic “improvement” of the results: 2 Trotter steps - analyzing differences

Heuristic “improvement” of the results: 3 Trotter steps - analyzing differences

Unitary evolution. General scheme. I. Representation of Hamiltonian through spin variables. Correspondence between the degrees of freedom of modeled system and qubits of the chip. II. Preparation of the initial state in real quantum device III. Simulation of unitary (free) evolution in real quantum device using evolution operator and its Trotterization via discrete operations

Intermediate summary - The dependence of the dynamics on the initial state is reproduced -Entanglement, quantum interference, excitation blockade -Very few Trotter steps can be implemented due to the errors of two-qubit gates -Results of the modeling can be improved to some extent using error handling and data processing

Initial state of the system – entangled “bath” tunable phase parameter. Dynamics of the central spin can be suppressed due to the negative quantum interference of contributions from three qubits. Two-step encoding via the quantum teleportation

Full circuit for encoding 3PES SWAP

Full circuit for the whole algorithm

Dark and bright states: quantum superpositions of two-particle entangled states Entanglement in the bath and quantum interference effects block excitation transfer to the center Three-particle entangled state: Population of central particle experimenttheory

Encoding

Full circuit for the whole algorithm

Rabi oscillations of collective spin Collective (cooperative) behavior of spins is reproduced Unexcited bath: Population of central particle experimenttheory

III. Transverse-field Ising model and 16-qubit IBM device Ising model in a transverse field – simplest playground to study far-from-equilibrium dynamics and thermalization. Non-stochastic model. initial state

8-spin Ising chain after 1 Trotter step: experiment vs theory

16-spin Ising ladder after 1 Trotter step: experiment vs theory Errors are enhanced but qualitative behavior is correct

Heuristic “improvement” of the results: 1 Trotter step Analysis of variations (properly normalized)

Heuristic “improvement” of the results: 2 Trotter steps Analysis of variations (properly normalized)

Single-qubit gates can be implemented with the high fidelity Two-qubit gates are very problematic Typical error in superconducting realization is of the order of 1%. Estimation of total error for spin (!) models Gate errors To have a error of the order of 1 % after 10 Trotter steps, CNOT error must be 10^(-4) Increase of Trotter number – decrease of (mathematical) Trotterization error, but increase of (physical) errors of the device

Superconducting quantum computers are perspective for simulation of far-from-equilibrium dynamics of various spin models in 1D and 2D. Addressability of qubits – various initial states. Flexibility. Tunability. Proof-of-principle demonstration with IBM quantum machines. Errors of two-qubit gates are too high. Must be reduced at least for one order in magnitude. Scaling is less important than reduction of gate errors? Tens of qubits + reduced errors = quantum supremacy? Dynamics of spin models – nearest application of QC for quantum modeling. Summary