Estimation… with quantum technology… using machine learning… on the blockchain

A snarky academic joke which might actually be interesting (but still a snarky joke).

Abstract

A device verification protocol using quantum technology, machine learning, and blockchain is outlined. The self-learning protocol, SKYNET, uses quantum resources to adaptively come to know itself. The data integrity is guaranteed with blockchain technology using the FelixBlochChain.

Introduction

You may have a problem. Maybe you’re interested in leveraging the new economy to maximize your B2B ROI in the mission-critical logistic sector. Maybe, like some of the administration at an unnamed university, you like to annoy your faculty with bullshit about innovation mindshare in the enterprise market. Or, maybe like me, you’d like to solve the problem of verifying the operation of a physical device. Whatever your problem, you know about the new tech hype: quantum, machine learning, and blockchain. Could one of these solve your problem? Could you really impress your boss by suggesting the use of one of these buzzwords? Yes. Yes, you can.

Here I will solve my problem using all the hype. This is the ultimate evolution of disruptive tech. Synergy of quantum and machine learning is already a hot topic1. But this is all in-the-box. Now maybe you thought I was going outside-the-box to quantum agent-based learning or quantum artificial intelligence—but, no! We go even deeper, looking into the box that was outside the box—the meta-box, as it were. This is where quantum self-learning sits. Self-learning is protocol wherein the quantum device itself comes to learn its own description. The protocol is called Self Knowing Yielding Nearly Extremal Targets (SKYNET). If that was hard to follow, it is depicted below.

hypebox
Inside the box is where the low hanging fruit lies—pip install tensorflow type stuff. Outside the box is true quantum learning, where a “quantum agent” lives. But even further outside-the-meta-box is this work, quantum self-learning—SKYNET.

Blockchain is the technology behind bitcoin2 and many internet scams. The core protocol was quickly realised to be applicable beyond digital currency and has been suggested to solve problems in health, logistics, bananas, and more. Here I introduce FelixBlochChain—a data ledger which stores runs of experimental outcomes (transactions) in blocks. The data chain is an immutable database and can easily be delocalised. As a way to solve the data integrity problem, this could be one of the few legitimate, non-scammy uses of blockchain. So, if you want to give me money for that, consider this the whitepaper.

Problem

 

99probs
Above: the conceptual problem. Below: the problem cast in its purest form using the formalism of quantum mechanics.

The problem is succinctly described above. Naively, it seems we desire a description of an unknown process. A complete description of such a process using traditional means is known as quantum process tomography in the physics community3. However, by applying some higher-order thinking, the envelope can be pushed and a quantum solution can be sought. Quantum process tomography is data-intensive and not scalable afterall.

The solution proposed is shown below. The paradigm shift is a reverse-datafication which breaks through the clutter of the data-overloaded quantum process tomography.

fuckyeahquantum
The proposed quantum-centric approach, called self-learning, wherein the device itself learns to know itself. Whoa. 

It might seem like performing a measurement of \{|\psi\rangle\!\langle \psi|, \mathbb I - |\psi\rangle\!\langle \psi|\} is the correct choice since this would certainly produce a deterministic outcome when V = U. However, there are many other unitaries which would do the same for a fixed choice of |\psi\rangle. One solution is to turn to repeating the experiment many times with a complete set of input states. However, this gets us nearly back to quantum process tomography—killing any advantage that might have been had with our quantum resource.

Solution

quantumintensifies
Schematic of the self-learning protocol, SKYNET. Notice me, Senpai!

This is addressed by drawing inspiration from ancilla-assisted quantum process tomography4. This is depicted above. Now the naive looking measurement, \{|\mathbb I\rangle\!\langle\mathbb I |, \mathbb I - |\mathbb I\rangle\!\langle \mathbb I|\}, is a viable choice as

|\langle\mathbb I |V^\dagger U \otimes \mathbb I |\mathbb I\rangle|^2 = |\langle V | U\rangle|^2,

where |U\rangle = U\otimes \mathbb I |\mathbb I\rangle. This is exactly the entanglement fidelity or channel fidelity5. Now, we have |\langle V | U\rangle| = 1 \Leftrightarrow U = V, and we’re in business.

Though |\langle V | U\rangle| is not accessible directly, it can be approximated with the estimator P(V) = \frac{n}{N}, where N is the number of trials and n is the number of successes. Clearly, \mathbb E[P(V)] = |\langle V | U\rangle|^2.

Thus, we are left with the following optimisation problem:
\min_{V} \mathbb E[P(V)] \label{eq:opt},

subject to V^\dagger V= \mathbb I. This is exactly the type of problem suitable for the gradient-free cousin of stochastic gradient ascent (of deep learning fame), called simultaneous perturbation stochastic approximation6. I’ll skip to the conclusion and give you the protocol. Each epoch consists of two experiments and a update rule:

V_{k+1} = V_{k} + \frac12\alpha_k \beta_k^{-1} (P(V+\beta_k \triangle_k) - P(V-\beta_k \triangle_k))\triangle_k.

Here V_0 is some arbitrary starting unitary (I chose \mathbb I). The gain sequences \alpha_k, \beta_k are chosen as prescribed by Spall6. The main advantage of this protocol is \triangle_k, which is a random direction in unitary-space. Each epoch, a random direction is chosen which guarantees an unbiased estimation of the gradient and avoids all the measurements necessary to estimation the exact gradient. As applied to the estimation of quantum gates, this can be seen as a generalisation of Self-guided quantum tomography7 beyond pure quantum states.

To ensure integrity of the data—to make sure I’m not lying, fudging the data, p-hacking, or post-selecting—a blochchain-based solution is implemented. In analogy with the original bitcoin proposal, each experimental datum is a transaction. After a set number of epochs, a block is added to the datachain. Since this is not implemented in a peer-to-peer network, I have the datachain—called FelixBlochChain—tweet the block hashes at @FelixBlochChain. This provides a timestamp and validation that the data taken was that used to produce the final result.

Results

results
SKYNET finds a description of its own process. Each N is a different number of bits per epoch. The shaded region is the interquartile range over 100 trials using a randomly selected “true” gate. The solid black lines are fits which suggest the expected 1/\sqrt{N} performance.

Speaking of final result, it seems SKYNET works quite well, as shown above. There is still much to do—but now that SKYNET is online, maybe that’s the least of our worries. In any case, go download the source8 and have fun!

Acknowledgements

The author thanks the quantum technology start-up community for inspiring this work. I probably shouldn’t say this was financially supported by ARC DE170100421.


  1. V. Dunjko and H. J. Briegel, Machine learning and artificial intelligence in the quantum domain, arXiv:1709.02779 (2017)
  2. N. Satoshi, Bitcoin: A peer-to-peer electronic cash system, (2008), bitcoin.org. 
  3. I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997)
  4. J. B. Altepeter, D. Branning, E. Jerey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomography, Phys. Rev. Lett. 90, 193601 (2003)
  5. B. Schumacher, Sending quantum entanglement through noisy channels, arXiv:quant-ph/9604023 (1996)
  6. J. C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE Transactions on Automatic Control 37, 332 (1992)
  7. C. Ferrie, Self-guided quantum tomography, Physical Review Letters 113, 190404 (2014)
  8. The source code for this work is available at https://gist.github.com/csferrie/1414515793de359744712c07584c6990

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