Learning Adaptive Quantum State Tomography with Neural Networks & Differentiable Programming

Stanislav Fort Stanford University (prev. Google Research)

[email protected] stanford.edu/~sfort1/ @stanislavfort

Primarily based on arXiv 1812.06693

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Primarily based on arXiv 1812.06693, in review

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA AI, ML, DL & differentiable programming

● Artiﬁcial intelligence: The science of making machines smart ● Machine learning: Machines getting smart from data ● Deep learning: Machines getting smart from data using deep neural networks as functional approximators ● Differentiable programming: Taking partial derivatives through programs, not restricted to deep neural networks as functions

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA AI, ML, DL & differentiable programming

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Differentiable programming Heat equation example ∂ Bridging ML & scientiﬁc computing

∂ temperature ﬁeld

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Differentiable programming Quantum games example ∂

∂ quantum game strategy

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Writing down a solution vs learning a solution Bubble sort (explicit) Image classiﬁcation (learned) f( )= “tortoise”

Translation (much better when learned)

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Writing down a solution vs learning a solution

Even solving symbolic math

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Machine learning Learning to get faster (& better) heuristics

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Network & data structure How to induce the correct learning prior?

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst ? ? ? ? ?

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst ? ? ? ? ?

(Adaptivity)

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst ? ? ? ? ?

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst

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Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst ? ? ?? ????? ?? ?? ??

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst N ? ? ?? ?? ? ? ?????? ?? ?? ??

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography Pure states ﬁrst N ? ??? ???? ? ?????? ?? ? ??

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Quantum state tomography What is difﬁcult? N ? ??? ???? ? ?????? ?? ? ??

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Pure states → density matrices

Pure states Density matrices

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA What do we care about?

1) How precisely can you reconstruct the unknown state, given N copies of the unknown state?

2) How much compute does it take?

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Regular Quantum State Tomography

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Regular Quantum State

Stage 1: Take measurementsTomography

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Regular Quantum State

TomographyStage 2: Process them

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Regular Quantum State Tomography Problem: Measurements expensive

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Adaptive Quantum State Tomography Ferenc Huszár, Neil M. T. Houlsby. Adaptive Bayesian Quantum Tomography, arXiv 1107.0895

Stage 1: Take measurements up to t

Stage 2: Determine the next POVM

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Adaptive Quantum State Tomography Particle bank maintaining and update

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Adaptive Quantum State Tomography Particle bank resampling

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Adaptive Quantum State Tomography

Problem: Computationally expensive

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Neural Adaptive Quantum State Tomography (NA-QST) Our approach Yihui Quek*, Stanislav Fort*, Hui Khoon Ng. Adaptive Quantum State Tomography with Neural Networks, arXiv 1812.06693 Advantages: - Parametrized state ansatz is not required - Exponential computational speedup - Learned directly from simulated data - Can retrain within hours - Any POVM types - Different noise models - Learned distance measure, update rule etc. Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Neural Adaptive Quantum State Tomography (NA-QST) Custom recurrent architecture - off the shelf doesn’t work

Train: Differentiable quantum simulator provides measurement results

Test: Experimenter provides measurement results

● No explicit Bayesian interpretation - weights are just weights ● Automatically learned NN similarity metric for resampling and weight updates ● Resampling is fast ● End-to-end training minimizing arbitrary human choices

1) Single-qubit POVM type: 2 (basis POVM), 3, 4 (SIC), and 6 (Pauli) legs per qubit subspace

2) POVM Adaptivity: Adaptive or random measurements? What are the beneﬁts?

3) 3) Reconstruction algorithm: Standard QST, ABQT, or our NA-QST

Number of measurements

Reconstruction error

NA-QST is:

1) Equal in reconstruction accuracy

NA-QST is:

1) Equal in reconstruction accuracy

2) Orders of magnitude faster!

NA-QST is:

1) Equal in reconstruction accuracy

2) Orders of magnitude faster!

3) Time complexity scales better

ABQT: polynomial time scaling

NA-QST: logarithmic time scaling

Conclusions:

- 2 legs (basis): Adaptivity helps a lot

- 3 legs: Adaptive helps slightly

- 4 legs and above (informationally complete): Adaptivity does not make any difference

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA Takeaways and thank you! We designed, implemented, and tested an end-to-end trainable, deep learning powered algorithm called Neural Adaptive Quantum State Tomography (NA-QST)

Primarily based on

arXiv 1812.06693 NA-QST is:

1) Fast to train (hours on a laptop) 2) Very fast to run (poly → log) 3) Accurate in reconstruction (~SOTA) 4) Flexible (noise, different POVMs etc)

Future: Retraining for downstream products involving the density matrix

Stanislav Fort | Adaptive Quantum State Tomography with NNs and differentiable programming | 19 February 2020 | Stanford, USA