A Faulttolerant Architecture for Quantum Hamiltonian Simulation Guoming

A Fault-tolerant Architecture for Quantum Hamiltonian Simulation Guoming Wang Oleg Khainovski

Background w Quantum Mechanical Systems are everywhere ª nuclear reaction, chemical molecules, superconductor, DNA, . . . w Quantum system’s states are vectors of exponential length in the number of particles it contains Large-scale quantum systems are hard to simulate on classical computers w Quantum simulation is the original motivation for building a quantum computer [suggested by Richard Feynman in 1982] w Efficient simulation of quantum systems is perhaps the most important application of quantum computers

Hamiltonian w Shrödinger’s Equation w H(t) -- Hamiltonian ª a matrix that represents total energy of the system ª usually a sum of local terms 2 D Ising Model

Hamiltonian Simulation w For time-independent Hamiltonian w Time-independent Hamiltonian Simulation Problem ª Given the description of a Hamiltonian H and a time t, build a polynomial-size quantum circuit that approximates the unitary transformation

Quantum CAD Flow Quantum CAD flow Circuit Specification Error Correction Synthesis Encoded Circuit Datapath Synthesis Mapping FU layout & Unmapped Circuit Mapped Circuit Too failures? many Verification Success Probability Full Layout Routing Area & Latency

Our Work w Studied the architecture for Hamiltonian Simulation using Quantum CAD flow ª A software that, given a Hamiltonian Simulation problem, generates and optimizes the solution circuit, and then feeds it into the CAD flow ª Optimizations to the Error Correction Synthesis, Datapath Synthesis and Mapping stages of the CAD flow Quantum CAD flow Hamiltonian Simulation Problem Software Circuit Specification Error Correction Synthesis Encoded Circuit Datapath Synthesis FU layout & Unmapped Circuit Mapping Mapped Circuit Too many failures? Verification Success Probability Full Layout Routing Area & Latency

How to Simulate Hamiltonians w Basic Principle w Outline

How to Simulate Hamiltonians w Each local term only acts on few number of qubits, and can be implemented by a relatively small circuit Use Solovay-Kitaev algorithm to find a short sequence of basic instructions

Optimization by Layering w Observation: ª The ordering of local terms affects the parallelism of the resulting circuit ª define “layers” of local terms --- all local terms in the same layer are independent ª Term-by-Term Layer-by-Layer ª use a greedy algorithm to find layers

Optimization by Layering w Particularly good for Ising Model ª number of layers independent of number of qubits

Standard Error Correction w Quantum Error Correction ª Two Stages: w Correct X (bit flip) error w Correct Z (phase flip) error w Standard Error Correction: place correction after every gate ª too expensive (>90% physical operations) ª more gates and movements more errors?

Selective Error Correction w Selective Error Correction ª place fewer corrections on the Critical Error Path w define an Error Distance Threshold w CNOT propagates the input errors ª reduced gate count & satisfactory success probability

Selective XZ Error Correction w Observation: ª X (Bit flip) and Z (Phase flip) errors have different behaviors ª Correct them separately further reduce gate count

Datapath Organizations w Qalypso ª variable sized compute and memory regions, ancilla generators, teleportation network ª determined based on an analysis of the given circuit or user’s choice

Qalypso+ w Idea: reduce the number of expensive long-range teleportation communications ª analyze the given circuit, construct a graph whose vertices are data qubits and edges are their interactions ª find a relatively small and balanced cut of this graph ª each part of data qubits are assigned to a particular compute region as its “favorite” qubits

Mapping w For every gate in the program order, decide which functional unit is used to execute it and hence how to move the data qubits ª evaluate every functional unit to find the best ª if a compute region does not like one of the input qubits, then all the functional units it contains will get a penalty w By this rule, every data qubit lives in a particular compute region most of the time and moves out only when necessary w This partitioning and mapping strategy is particularly good for Hamiltonian Simulation Circuits ª particles normally interact only with its neighbors

Metric for Probabilistic Computation w Metric: Area-Delay-to-Correct-Result (ADCR) w For ADCR, lower is better.

Experimental Results w Effect of Layering for Ising Model

Experimental Results w Effect of Layering for General Hamiltonian

Experimental Results w Comparison of Datapaths for General Hamiltonian

Experimental Results w Comparison of Datapaths for Random Circuit

Experimental Results w Comparison of Error Correction Schemes

Experimental Results w Effect of Error Distance Threshold

Experimental Results w Overall for General Hamiltonian

Experimental Results w Overall for General Hamiltonian

Future Direction w Further Improvement to Hamiltonian Simulation ª Better Layering? ª Better Partitioning and Mapping? ª Other aspects: Routing? Ancilla factory? w Extension to Time-dependent Hamiltonian Simulation ª Adiabatic algorithms? w Application: studying materials? solving linear systems?