the born supremacy: quantum advantage and training of an ... · quantum computers as samplers...
TRANSCRIPT
![Page 1: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/1.jpg)
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
Brian Coyle, Daniel Mills, Vincent Danos, Elham Kashefi
arXiv:1904.02214
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
![Page 3: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/3.jpg)
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The BORN Supremacy
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The BORN Supremacy
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The BORN Supremacy
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
![Page 8: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/8.jpg)
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
![Page 9: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/9.jpg)
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
![Page 12: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/12.jpg)
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
Generative
Modeling
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The Task: Generative Modeling
“Generative Modeling is the use of Artificial Intelligence, statistics and probability in applications to produce a representation or abstraction of observed phenomena or target variables that can be calculated from observations.”
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The Task: Generative Modeling
“Generative Modeling is the use of Artificial Intelligence, statistics and probability in applications to produce a representation or abstraction of observed phenomena or target variables that can be calculated from observations.”
Untrained Generative
Model
Train the model using ‘samples’
![Page 15: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/15.jpg)
The Task: Generative Modeling
“Generative Modeling is the use of Artificial Intelligence, statistics and probability in applications to produce a representation or abstraction of observed phenomena or target variables that can be calculated from observations.”
Untrained Generative
Model
Trained Generative
Model
Train the model using ‘samples’
![Page 16: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/16.jpg)
The Task: Generative Modeling
“Generative Modeling is the use of Artificial Intelligence, statistics and probability in applications to produce a representation or abstraction of observed phenomena or target variables that can be calculated from observations.”
Untrained Generative
Model
Trained Generative
Model
Train the model using ‘samples’Trained Model will (approximately) generate a ‘new’ cat image
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Quantum Computers as Samplers
Classical - Neural Network or other
Quantum - Parameterised Quantum Circuit (PQC) arXiv:1906.07682
Generative Model
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Quantum Computers as Samplers
● Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642● Quantum Boltzmann Machine Phys. Rev. X 8, 021050
○ Using annealing to prepare thermal state to sample from.● Gate based Quantum Boltzmann Machine arXiv:1712.05304
○ Use QAOA to prepare approximate thermal state.● Born Machine npj QI 5:45, Phys. Rev. A 98, 062324, ...
○ A ‘new’ model - generates statistics directly from Born rule of Quantum Mechanics
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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Ising Born Machine
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Ising Born Machine
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Ising Born Machine
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Ising Born Machine
Trainable Parameters
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Ising Born Machine
Recover IQP (Instantaneous Quantum Polytime) circuits.
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Ising Born Machine
Recover shallowest depth (p=1) version of QAOA circuits.
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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Gradient Based Training
1 - Sample from model.
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Gradient Based Training
2 - Evaluate loss & gradient
1 - Sample from model.
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Gradient Based Training
3 - Update parameters by gradient based descent
2 - Evaluate loss & gradient
1 - Sample from model.
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Choosing a Cost Function
Computing the loss function is a means of checking how well we are doing - comparing the data and the instantaneous model distributions.
How do we compare two probability distributions? - This is hard.
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Choosing a Cost Function
Computing the loss function is a means of checking how well we are doing - comparing the data and the instantaneous model distributions.
How do we compare two probability distributions? - This is hard.
![Page 34: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/34.jpg)
Choosing a Cost Function
Computing the loss function is a means of checking how well we are doing - comparing the data and the instantaneous model distributions.
How do we compare two probability distributions? - This is hard.
![Page 35: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/35.jpg)
Choosing a Cost Function
We need a loss function which is ideally:
● Easily computable (in terms of sample + computational complexity)● Relatively ‘powerful’ (should be sensitive to differences in the distributions)
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The Benchmark - Total Variation Distance
Why? It’s the notion that typically goes with quantum supremacy experiments:
● IQP: Phys. Rev. Lett. 117, 080501- Assume a conjecture about the hardness of computing the Ising partition function. If it is possible to classically sample from the output probability distribution of any IQP circuit C in polynomial time, up to an error of 1/384 in TV, then there is a BPP^NP algorithm to solve any problem in P^#P . Hence the Polynomial Hierarchy would collapse to its third level.
Model Data
![Page 37: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/37.jpg)
The Benchmark - Total Variation Distance
Why? It’s the notion that typically goes with quantum supremacy experiments:
● IQP: Phys. Rev. Lett. 117, 080501- Assume a conjecture about the hardness of computing the Ising partition function. If it is possible to classically sample from the output probability distribution of any IQP circuit C in polynomial time, up to an error of 1/384 in TV, then there is a BPP^NP algorithm to solve any problem in P^#P . Hence the Polynomial Hierarchy would collapse to its third level.
Model Data
Powerful
![Page 38: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/38.jpg)
The Benchmark - Total Variation Distance
Why? It’s the notion that typically goes with quantum supremacy experiments:
● IQP: Phys. Rev. Lett. 117, 080501- Assume a conjecture about the hardness of computing the Ising partition function. If it is possible to classically sample from the output probability distribution of any IQP circuit C in polynomial time, up to an error of 1/384 in TV, then there is a BPP^NP algorithm to solve any problem in P^#P . Hence the Polynomial Hierarchy would collapse to its third level.
Model Data
PowerfulHard to compute
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Previous Training - Maximum Mean
DiscrepancyLiu & Wang: Phys. Rev. A 98, 062324, Gretton et.al.: JMLR 13 (2012) 723-773
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Previous Training - Maximum Mean
Discrepancy
The MMD is very efficient to compute. It has quadratic sample complexity independent of the size of the underlying space arXiv:0901.2698 :
But, it lower bounds Total Variation: On Choosing and Bounding Probability Metrics
So, minimising MMD, does not necessarily do a good job of minimising TV. Can we minimise an efficient upper bound instead?
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Wasserstein & Optimal Transport?
The Wasserstein metric is related to Optimal Transport (Villani, 2009: Optimal transport, old and new. ) - a way to compare distributions by determining how to ‘transport’ one into the other
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Wasserstein & Optimal Transport?
The Wasserstein metric is related to Optimal Transport (Villani, 2009: Optimal transport, old and new. ) - a way to compare distributions by determining how to ‘transport’ one into the other
![Page 43: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/43.jpg)
Wasserstein & Optimal Transport?
The Wasserstein metric is related to Optimal Transport (Villani, 2009: Optimal transport, old and new. ) - a way to compare distributions by determining how to ‘transport’ one into the other
![Page 44: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/44.jpg)
Wasserstein & Optimal Transport?
The Wasserstein metric is related to Optimal Transport (Villani, 2009: Optimal transport, old and new. ) - a way to compare distributions by determining how to ‘transport’ one into the other
![Page 45: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/45.jpg)
Wasserstein & Optimal Transport?
The Wasserstein metric is related to Optimal Transport (Villani, 2009: Optimal transport, old and new. ) - a way to compare distributions by determining how to ‘transport’ one into the other
OT ‘cost’ - Distance between points in p, q. - Take to be Hamming distance between bin strings.
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Computation & Power
OT is hard to compute though arXiv:0901.2698...
But it does upper bound TV: On Choosing and Bounding Probability Metrics
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Computation & Power
OT is hard to compute though arXiv:0901.2698...
But it does upper bound TV: On Choosing and Bounding Probability Metrics
![Page 48: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/48.jpg)
Sinkhorn Divergence
Let’s add a regularisation term to the optimal transport distance… While we’re at it, let’s also add symmetric terms to remove bias... => Sinkhorn Divergence Entropy 2017, 19(2), 47, arXiv:1810.08278, arXiv:1706.00292
Regularised by the entropy term (KL divergence). Sinkhorn divergence interpolates between MMD and unregularised optimal transport (as a function of regulariser, 𝝐) - gradient has same form as MMD.
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Sinkhorn Divergence
Can be efficient arXiv:1810.02733
(With prob. 1-δ)
Can also be powerful:
Un-regularised Optimal transport Regularised Optimal transport
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Sinkhorn vs MMD
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
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Quantum Advantage of Training
“Quantum supremacy is the potential ability of quantum computing devices to solve problems that classical computers practically cannot.” John Preskill
The simplest example of such a problem is a sampling problem (or at least that we have evidence for)
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Quantum Advantage of Training
Part 1 - Hardness of Simulation:
● Hardness of simulating the IBM (Ising Born machine) can be retained through training by enforcing parameter updates in a particular way.
BUT: This doesn’t say the model is able to actually outperform all classical algorithms in a learning task - Hardness of simulation does not imply usefulness!
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Quantum Advantage of Training
Part 2 - Advantage in learning (??):
● Can we find examples which can be reached by quantum models, but cannot by any classical models?
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The Order of Definitions
Supremacy
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The Order of Definitions
Learning Supremacy
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The Order of Definitions
Learning Supremacy
Generator
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The Order of Definitions
Learning Supremacy
Quantum Learning
Supremacy!!
Generator
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The Order of Definitions
Learning Supremacy
Quantum Learning
Supremacy!!
Generator
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Supremacy
Supremacy null hypothesis:
The output of this computation was arrived at by a classical computer
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The Order of Definitions
Learning Supremacy
Quantum Learning
Supremacy!!
Generator
![Page 63: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/63.jpg)
Generator
Generator
Collection of probability distributions
Samples from
distribution in collection
![Page 64: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/64.jpg)
Generator
An ‘Approximate Generator’, adapted from Kearns '94 - On the learnability of discrete distributions:
![Page 65: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/65.jpg)
Learning
Learning algorithm
Generator for distribution D’ which is close to D
![Page 66: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/66.jpg)
The Order of Definitions
Learning Supremacy
Quantum Learning
Supremacy!!
Generator
![Page 67: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/67.jpg)
Quantum Learning Supremacy
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Conclusions
● We defined the Ising Born Machine.● We used new gradient training methods - Sinkhorn Divergence - which is ‘stronger’ than the MMD,
but efficiently computable.● Quantum Advantage - By connecting to IQP and QAOA, the model is hard to sample from and can
remain hard during training. We defined a framework for a provable advantage for generative modelling, potentially in the near term.
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Conclusions
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine: arXiv:1904.02214
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Conclusions
The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine: arXiv:1904.02214
![Page 71: The Born Supremacy: Quantum Advantage and Training of an ... · Quantum Computers as Samplers Quantum Inspired Training of Boltzmann Machines arXiv:1507.02642 Quantum Boltzmann Machine](https://reader034.vdocuments.net/reader034/viewer/2022050201/5f548bce9d248524475fc888/html5/thumbnails/71.jpg)
Additional Plot