using Low-Depth Quantum Circuits Guillaume Verdon Co-Founder & Chief Scientific Officer

Transcription

1 Training Neural Networks using Low-Depth Quantum Circuits QC Guillaume Verdon Co-Founder & Chief Scientific Officer

2 Overview

3 Classical Boltzmann machines? Energy-based model X X E(z) = J jk z j z k B j z j hjki j visibles hiddens Use thermal equilibrium statistics to mimic data P (z) = e E(z) P z e E(z)

4 Classical Boltzmann machines? Goal: reduce visibles relative entropy to data X Pvis (x) P data (x) log P data (x) x2data {z } D KL (P data P vis ) Need to sample from thermal (Gibbs) distribution for training Use Monte Carlo techniques Or... quantum computers! Restricted Boltzmann Machine (RBM)

5 Quantum Boltzmann machines? Energy function à Hamiltonian operator X Ĥ = J jk Ẑ j Ẑ k B j Ẑ j X hjki j visibles hiddens For Gibbs sampling: need to simulate quantum thermalization Quantum Analog Quantum Digital Quantum Approximate H = j H j, H j = C 2

6 Quantum analog Gibbs sampling Quantum Annealers Physical system enacting thermalization Physics of chip are direct analogy of algorithm Caveats: Physical temperature Connectivity issues Low-quality (flux) qubits Less flexible/embedding problems Quantum Analog: physical evolution governed by the physics of the chip

7 Quantum simulated thermalization with circuits Theoretically: Quantum simulated to arbitrary accuracy, given enough gates No connectivity problems (virtualized physics) Caveats: Would need full fault-tolerance >10 9 gates Near-term? Could we do something similar but tailored to Noisy Intermediate Scale Quantum Devices? Yes. Quantum Simulation: evolution decomposed into small pulses approximating evolution path

8 Quantum-classical hybrid thermalization Hybrid Variational approach Fixed number of quantum pulses Classically optimize pulse lengths CPU+QPU share optimization load Quantum Approximate Optimization Algorithm (QAOA) inspired from Adiabatic QC Proven quantum supremacy Near-term implementable Quantum-Classical Hybrid: fixed number of pulses, classically variationally optimized to mimimize distance to target state

9 Technical Background

10 Training quantum Boltzmann machines e H H = X hjki J jk Z j Z k X j B j Z j Network Hamiltonian D( data k vis ) Network parameters Want to minimize Gradient descent? D( data V x = k log vis xihx ) v Too hard. Minimize upper bound instead

11 Bound-based update rule H = X hjki J jk Z j Z k X B j Z j j Minimize KL Minimize upper-bound 1 via update rule: Positive phase Gibbs sampling needed for each term Negative phase V x = log xihx v [1] Amin, M. H., Andriyash, E., Rolfe, J., Kulchytskyy, B., & Melko, R., Quantum Boltzmann Machine, [arxiv: ]

12 Example: updating weights J jk j k Jjk H = Z jz k Z = "ih" #ih# D = {0010, 1100,...} = {""#", ""##,...} J jk = 1 D Measure clamped & unclamped thermal correlations X x2d hz j Z k i clamped, x hz j Z k i unclamped V x = log xihx v

13 Example: updating weights J Jjk H = Z j Z k Dataset D = {0010, 1100,...} ( = {""#", ""##,...} " k " j # k " j + " k " j Pos. phase # Neg. phase # + J jk = 1 D Clamped Gibbs. ( Unclamped Gibbs

14 Classical Alternatives to Clamped Sampling For RBM s: Feedforward data points For more general BM s Markov Chain Monte Carlo sampling Other sampling methods (QMC etc.) V x = log xihx v

15 Quantum-Classical Variational Algorithms Goal: find minimizing Parameterize a family of ansatz circuits U( 1 ) U( 2 ) U( 3 ) U( 4 ) U( 5 ) U( 6 ) Loop: CPU QPU until min reached hhi ~ V x = log xihx v

16 Quantum Approximate Optimization Algorithm Hybrid variational algo inspired from Adiabatic QC Mixer: H M = P j X j Cost: H C = P hjki J jkz j Z k CPU QPU Analog adiabatic Trotterized Simulation QAOA

17 Our algorithm

18 Quantum Approximate Thermalization Say we want to sample from BM Gibbs state e H bm where H bm = P hjki J jkz j Z k Pj B jz j Initiate system in easy thermal state or hh C i Use QAOA to min with Cost: H C H bm Mixer: H M = P j X j Brings our state closer to thermal (Gibbs) state Free energy F = hh C i 1 S = D( k ) Relative entr.

19 Quantum Approximate Boltzmann Machines QABoM Train (semi-)restricted BM using bound-based rule X J jk = 1 D hz j Z k i clamped, x hz j Z k i unclamped x2d Use Quantum Approximate Thermalization for Unclamped + clamped Gibbs sampling Clamping can be regular or quantum randomized j J jk k For RBM case Can do positive phase + inference classically

20 ( Quantum Randomized Clamping QRC for QABoM J jk = 1 D Randomize clamping during QAOA faster training AND better performance ( P x2d hz jz k i clamped, x hz j Z k i unclamped 1 D ( " " j # " + " " j # # +. k k j k ( " " " " + D # # " # = 1 1 D +. P x2d xihx j k j k

21 QABoM paper results [arxiv: ] Training an RBM with noisy(depolarizing) circuits using Rigetti Forest QVM

22 It s coding time!

23 Code walkthrough.com/michaelbroughton/qabom

24 Importing dependencies Setting up pyquil Importing both QAOA and VQE

25 Setting up the qrbm class Network setup Temp and QAOA steps

26 Setting up the qrbm class Use VQE for expect. values Setting up random initial network parameters

27 Setting up the unclamped QAOA sampling Define hidden/visible indices

28 Setting up the unclamped QAOA sampling Mixer Hamiltonian H M = P j X j Cost Hamiltonian H C = P hjki J jkz j Z k Pj B jz j

29 Setting up the unclamped QAOA sampling Prep thermal state using entanglement (for faster sim) Want: e P j Z j O j X e z j z j ihz j Prep each qubit as: z j 2{±1} p 2 cosh( ) X e z zi S zi E R x ( ) z2{1, 1} +

30 Setting up the unclamped QAOA sampling Put QAOA ingredients together Set optimal QAOA angles output Use pyquil parametric program

31 Setting up the network training Option to mix quantum and classical RBM weight updates Find optimal QAOA given current weights

32 Setting up the network training Thermal correlation exp. value J jk = 1 D X x2d hz j Z k i clamped, x We compute only unclamped (neg phase) quantumly for demo hz j Z k i unclamped j J jk k

33 Setting up the network training Setting up classical positive phase via feedforward

34 Updating weights

35 Inference

36 Simple example: Hidden bit subspace Encode a random bit redundantly into 4 bits Let our qbm find the needle in the haystack Random coin flip Encdoding QABoM RBM decoding? Can it identify this hidden variable?

37 To Jupyter and beyond!

38 A brief intermission

39 About Everettian The Quantum Artificial Intelligence company. Advancing and democratizing bleeding-edge Quantum Machine Learning

40 The Quantum Valley Waterloo: the perfect place to build the quantum future Entanglement with Toronto s classical AI ecosystem

41 Join the team! Tom Lubowe CEO, Co-Founder - Experience in FinTech Product Management - Worked in Forex Execution, and Hedge Fund side of Finance Industry Guillaume Verdon CSO, Co-Founder - PhD student in Quantum Machine Learning at IQC & Perimeter Institute - MMath from IQC on Quantum Algorithms & Quantum Field Theory Mike Broughton CTO, Co-Founder - Computer Science at UWaterloo and Institute for Quantum Computing (IQC) - Early employee at Piinpoint, a Velocity & Y Combinator alumni company

42 Now back to the code

43 Outlook/Future work Hybrid quantum computers can train neural networks! Time to implement on a chip Possible improvements Variationally optimize temp/entropy Further achieve lower Free energy Extensions Apply to general BM Apply to deep BM Integrate into classical BM network

44 Thanks! Reach out: