Transforming the Digital Age Krysta Svore Quantum Architectures and - PowerPoint PPT Presentation

Quantum Computing: Transforming the Digital Age Krysta Svore Quantum Architectures and Computation (QuArC) Microsoft Research Quantum Optimization Workshop 2014

Antikythera mechanism (100 BC) Babbage’s Difference Engine (proposed 1822) ENIAC (1946) Sequoia (2012) Quantum (2025?) Thanks to Matthias Troyer

Is there anything we can’t solve on digital computers? 4

Some problems are hard to solve QMA-Hard QMA NP-Hard BQP NP P Ultimate goal: Develop quantum algorithms whose complexity lies in BQP\P

Quantum Magic: Interference interference pattern = source of quantum particles coherence Classical objects go either one way or the other. Quantum objects (electrons, photons) go both ways. Gives a quantum computation an inherent type of parallelism!

Quantum Magic: Qubits and Superposition 0 = | 〉 = ↓ 1 = = ↑ single atom single spin 𝒉 = 𝟏 ↓ = 𝟏 𝒇 = |𝟐〉 ↑ = |𝟐〉 𝜔 = 0 + 1 = + = ↓ + ↑ Information encoded in the state of a two-level quantum system

𝜔 = Thanks to Charlie Marcus

Input Output + + + +

Quantum Magic: Entanglement Nonlocal Correlations!

Quantum Magic: Entanglement 2*5 distinct amplitudes |𝜔 1 〉 𝑂 non-interacting qubits |𝜔 2 〉 |𝜔 0 〉 |𝜔 3 〉 |𝜔 4 〉 𝜔 𝑢𝑝𝑢𝑏𝑚 = 𝛽 0 0 + 𝛾 0 1 ⊗ 𝛽 1 0 + 𝛾 1 1 ⊗ ⋯ ⊗ 𝛽 𝑂−1 0 + 𝛾 𝑂−1 1 State of 𝑂 non-interacting qubits: ~ 𝑂 bits of info Thanks to Rob Schoelkopf

Quantum Magic: Entanglement 32 distinct amplitudes! General state of 𝑂 interacting qubits |𝜔 1 〉 |𝜔 2 〉 Simulating a 200-qubit interacting |𝜔 0 〉 system requires ~ 10 60 classical bits! |𝜔 3 〉 |𝜔 4 〉 𝜔 𝑢𝑝𝑢𝑏𝑚 = 𝑑 0 00 … 0 + 𝑑 1 00 … 1 + … 𝑑 2 𝑂−1 11 … 1 State of 𝑂 interacting qubits: ~ 2 𝑂 bits of info! Thanks to Rob Schoelkopf

Quantum Magic: What’s the catch? Need coherent Need strongly |𝜔 1 〉 control Interacting system |𝜔 2 〉 |𝜔 0 〉 |𝜔 3 〉 |𝜔 4 〉 Avoid interaction with Decoherence outside environment! and errors! Thanks to Rob Schoelkopf

Quantum Gates: Digital quantum computation Basic unit: qubit = unit vector Basic unit: bit = 0 or 1 𝛽 0 + 𝛾 1 Computing: logical operation Computing: unitary operation NOT NOT 0 → 1 𝛽 0 1 𝛾 = 𝛾 1 → 0 1 0 𝛽

Quantum Gates: Digital quantum computation Basic unit: qubit = unit vector Basic unit: bit = 0 or 1 𝛽 0 + 𝛾 1 Computing: logical operation Computing: unitary operation Description: truth table Description: unitary matrix A B Y 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 CNOT gate XOR gate

Quantum power unleashed: super-fast FFT # ops = N log N FFT Example: 1GB of data = 10 Billion ops Quantum # ops = log N FFT Example: 1GB of data = 27 ops (!!!)

Any other catches? No-cloning principle I/O limitations output measure + + Quantum information Input: preparing initial state can be costly cannot be copied Output: reading out a state is probabilistic

Quantum Algorithms Exist! Shor’s • Breaks RSA, elliptic curve Algorithm signatures, DSA, El-Gamal • Exponential speedups (1994) Solving Linear • Applications shown for electromagnetic wave scattering Systems of • Exponential speedups Equations (2010) • Simulate physical systems in a Quantum quantum mechanical device simulation (1982) • Exponential speedups

Cryptography

15 = ∎ × ∎

15 = 5 × 3

1387 = ∎ × ∎

1387 = 19 × 73

1807082088687 4048059516561 6440590556627 8102516769401 3491701270214 = ∎ × ∎ 5005666254024 4048387341127 5908123033717 8188796656318 2013214880557

1807082088687 3968599 4553449 4048059516561 9459597 8646735 4542901 9721884 6440590556627 Example: (n=2048 bits) 6112616 0368689 8102516769401 classically ~7x10 15 years 2883786 7274408 3491701270214 = × quantum ~100 seconds 0675764 8643563 5005666254024 4911281 0126320 4048387341127 5069600 0064832 5908123033717 5551572 9990445 8188796656318 2013214880557 43 99

Breaking RSA and elliptic curve signatures Classical: 1 2 3 log 𝑜 𝑃 exp 𝑜 3 Quantum: 𝑃 𝑜 2 log 𝑜

Machine learning

The Problem in Artificial Intelligence • How do we make computers that see , listen , and understand ? • Goal : Learn complex representations for tough AI problems • Challenges and Opportunities: • Terabytes of (unlabeled) web data, not just megabytes of limited data • No “silver bullet” approach to learning • Good new representations are introduced only every 5-10 years • Can we automatically learn representations at low and high levels? • Does quantum offer new representations? New training methods?

Deep networks learn complex representations object recognition Desired outputs 𝑚 1 𝑚 2 ⋯ 𝑚 𝑘 ⋯ 𝑚 𝐾 High-level features object properties 𝑤 1 𝑤 2 ⋯ 𝑤 𝑗 ⋯ 𝑤 𝐽 1 ℎ 1 ℎ 2 ⋯ ⋯ 1 ℎ 𝑘 ℎ 𝐾 Mid-level features textures 𝑤 1 𝑤 2 ⋯ 𝑤 𝑗 ⋯ 𝑤 𝐽 1 ℎ 1 ℎ 2 ⋯ ℎ 𝑘 ⋯ ℎ 𝐾 1 Difficult to specify exactly Low-level features edges 𝑤 1 𝑤 2 ⋯ 𝑤 𝑗 ⋯ 𝑤 𝐽 1 ℎ 1 ℎ 2 ⋯ ℎ 𝑘 ⋯ ℎ 𝐾 1 Deep networks learn these from data without Input pixels 𝑤 1 𝑤 2 ⋯ 𝑤 𝑗 ⋯ 𝑤 𝐽 1 explicit labels Analogy: layers of visual processing in the brain

What are the primary challenges in learning? • Desire: learn a complex representation (e.g., full Boltzmann machine) • Intractable to learn fully connected graph  poorer representation Can we learn a more complex representation on a quantum • Pretrain layers? computer? • Learn simpler graph with faster train time? • Desire: efficient computation of true gradient • Intractable to learn actual objective  poorer representation Can we learn the actual objective (true gradient) on a quantum computer? • Approximate the gradient? • Desire: training time close to linear in number of training examples • Slow training time  slower speed of innovation Can we speedup model training on a quantum computer? • Build a big hammer? • Look for algorithmic shortcuts?

Training RBM - Classical for each epoch //until convergence for i=1:N //each training vector CD(V_i, W) //CD given sample V_i and parameter vector W dLdW += dLdW //maintain running sum end W = W + (  /N) dLdW //take avg step end CD Time: # Epochs x # Training vectors x # Parameters ML Time: # Epochs x # Training vectors x (# Parameters) 2 x 2 |v| + |h|

Training RBM - Quantum for each epoch //until convergence for i=1:N //each training vector qML(V_i, W) //qML: Use q. computer tp CD(V_i, W) //CD given sample V_i and parameter vector W Approx. to sample P(v,h) dLdW += dLdW //maintain running sum end W = W + (  /N) dLdW //take avg step end !!! qML Time ~ # Epochs x # Training vectors x # Parameters qML Size (# qubits) for one call ~ |v| + |h| + K, K≤33

Quantum simulation

What does quantum simulation do? Physical Systems Quantum Chemistry Superconductor Physics Quantum Field Theory Computational Applications Emulating Quantum Computers Linear Algebra Differential Equations

Quantum simulation Particles can either be spinning clockwise (down) or counterclockwise (up) = 00000 ⋮ ⋮ ⋮ ⋮ ⋮ = 11111 There are 2 5 possible orientations in the quantum distribution. Cannot store this in memory for 100 particles.

Quantum Simulation for Quantum Chemistry Ultimate problem: Simulate molecular dynamics of larger systems or to higher accuracy Want to solve system exactly Current solution: 33% supercomputer usage dedicated to chemistry and materials modeling Requires simulation of exponential-size Hilbert space Limited to 50-70 spin-orbitals classically Quantum solution: Simulate molecular dynamics using quantum simulation Scales to 100s spin-orbitals using only 100s qubits Runtime recently reduced from 𝑃(𝑂 11 ) to 𝑃 𝑂 4 − 𝑃(𝑂 6 ) 37