Bespoke research in the fields of Quantum Machine Learning, Non-equilibrium thermodynamics, and the Physics of Information.

# Preprint: Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition

Gavin E. Crooks, arXiv:1905.13311

**Abstract**:

The parameter-shift rule is an approach to measuring gradients of quantum circuits with respect to their parameters, which does not require ancilla qubits or controlled operations. Here, I discuss applying this approach to a wider range of parameterize quantum gates by decomposing gates into a product of standard gates, each of which is parameter-shift rule differentiable.

# Preprint: Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope

Eric C. Peterson, Gavin E. Crooks, and Robert S. Smith, arXiv:1904.10541

[Full text | arXiv ]

**Abstract**:

For a native gate set which includes all single-qubit gates, we apply results from symplectic geometry to analyze the spaces of two-qubit programs accessible within a fixed number of gates. These techniques yield an explicit description of this subspace as a convex polytope, presented by a family of linear inequalities themselves accessible via a finite calculation. We completely describe this family of inequalities in a variety of familiar example cases, and as a consequence we highlight a certain member of the “XY-family” for which this subspace is particularly large, i.e., for which many two-qubit programs admit expression as low-depth circuits.

# Tech. Note: On the Weyl chamber of canonical non-local 2-qubit gates

PDF:

http://threeplusone.com/weyl

Source code:

https://github.com/gecrooks/on_weyl

The Weyl chamber of canonical non-local 2-qubit gates. Papercraft meets quantum computing. Print, cut, fold, and paste. (Should look like Fig. 4 of quant-ph/0209120)

A 2-qubit gate has 15 free parameters. But you can apply local 1-qubit gates before and after, which leaves a 15-4×3=3-parameter space of non-local gates. Once you remove a bunch of symmetries, you’re left with a tetrahedral chamber in which all your favorite 2-qubit gates live.

v3: Added ECP and Dagwood-Bumstead gates, and various other tweaks.

# QuantumFlow v0.8: Now with TensorFlow 2.0 support

QuantumFlow v0.8.0: Automatic differentiation of quantum circuits and SGD training of quantum networks. Now with TensorFlow 2.0 backend.

Install the latest tensorflow 2.0 alpha with

`> pip install -U --pre tensorflow`

and set the QUANTUMFLOW_BACKEND environment variable to tensorflow2.

`> QUANTUMFLOW_BACKEND=tensorflow2 make test`

# Tech. Note: Field Guide to Continuous Probability Distributions v0.12

Version: 0.12

A survey of probability distributions used to describe a single, continuous, unimodal, univariate random variable.

Whats New: Added Porter-Thomas, Epanechnikov, biweight, triweight, Libby-Novick, Gauss hypergeometric, confluent hypergeometric, Johnson~SU, and log-Cauchy distributions.

Full LaTeX source distributed on github: https://github.com/gecrooks/fieldguide

[ Full Text ]

# Tech. Note: On the Weyl chamber of canonical non-local 2-qubit gates

Tech. Note 012v1

PDF:

http://threeplusone.com/weyl

Source code:

https://github.com/gecrooks/on_weyl

The Weyl chamber of canonical non-local 2-qubit gates. Papercraft meets quantum computing. Print, cut, fold, and paste. (Should look like Fig. 4 of quant-ph/0209120)

A 2-qubit gate has 15 free parameters. But you can apply local 1-qubit gates before and after, which leaves a 15-4×3=3-parameter space of non-local gates. Once you remove a bunch of symmetries, you’re left with a tetrahedral chamber in which all your favorite 2-qubit gates live.

# Preprint: Performance of the Quantum Approximate Optimization Algorithm on the Maximum Cut Problem

Gavin E. Crooks, arXiv:1811.08419 (2018)

**Abstract**:

The Quantum Approximate Optimization Algorithm (QAOA) is a promising approach for programming a near-term gate-based hybrid quantum computer to find good approximate solutions of hard combinatorial problems. However, little is currently know about the capabilities of QAOA, or of the difficulty of the requisite parameters optimization. Here, we study the performance of QAOA on the MaxCut combinatorial optimization problem, optimizing the quantum circuits on a classical computer using automatic differentiation and stochastic gradient descent, using QuantumFlow, a quantum circuit simulator implemented with TensorFlow. Continue reading

# QuantumFlow: A Quantum Algorithms Development Toolkit v0.7

Announcing QuantumFlow, a python package that emulates a gate based quantum computer using modern optimized tensor libraries (numpy, TensorFlow, or torch). The TensorFlow backend can calculate the analytic gradient of a quantum circuit with respect to the circuitâ€™s parameters, and circuits can be optimized to perform a function using (stochastic) gradient descent. The torch backend can accelerate the quantum simulation using commodity classical GPUs.

Various other features include quantum circuits, circuit visualization, noisy quantum operations, gate decompositions, sundry metrics and measures, and an interface to Rigetti’s Forest infrastructure.

# Tech. Note 011 v0.3: On the Drazin inverse of the rate matrix

This technical note describes the Drazin pseudo-inverse, which is an under-appreciated mathematical gadget that has several interesting applications to non-equilibrium thermodynamics.

[ Full Text ]

# Tech. Note 009 v0.7: On Measures of Information and Entropy 009 v0.7

A brief overview of information measures on classical, discrete probability distributions. 009 v0.7 [ Full Text ]

# Preprint: Quantum Kitchen Sinks: An algorithm for machine learning on near-term quantum computers

C. M. Wilson, J. S. Otterbach, N. Tezak, Robert S. Smith, Gavin E. Crooks, and Marcus P. da Silva, arXiv:1806.08321 (2018)

[ Full text]

**Abstract**:

Noisy intermediate-scale quantum computing devices are an exciting platform for the exploration of the power of near-term quantum applications. Performing nontrivial tasks in such a framework requires a fundamentally different approach than what would be used on an error-corrected quantum computer. One such approach is to use hybrid algorithms, where problems are reduced to a parameterized quantum circuit that is often optimized in a classical feedback loop. Here we described one such hybrid algorithm for machine learning tasks by building upon the classical algorithm known as random kitchen sinks. Continue reading

# Preprint: Marginal and Conditional Second Laws of Thermodynamics (v2)

Gavin E. Crooks, Susanne Still arXiv:1611.04628

**Abstract**:

We consider the entropy production of a strongly coupled bipartite system. The total entropy production can be partitioned into various components, which we use to define local versions of the Second Law that are valid without the usual idealizations. The key insight is that the joint trajectory probability of interacting systems can be split into terms representing the dynamics of the individual systems without feedback.

# Article: Quantifying configuration-sampling error in Langevin simulations of complex molecular systems

Josh Fass, David A. Sivak, Gavin E. Crooks, Kyle A. Beauchamp, Benedict Leimkuhler, and John D. Chodera Entropy, 20(5):318 (2018).

**Abstract**:

While Langevin integrators are popular in the study of equilibrium properties of complex systems, it is challenging to estimate the timestep-induced discretization error: the degree to which the sampled phase-space or configuration-space probability density departs from the desired target density due to the use of a finite integration timestep. Sivak et al., introduced a convenient approach to approximating a natural measure of error between the sampled density and the target equilibrium density, the Kullback-Leibler (KL) divergence, in phase space, but did not specifically address the issue of configuration-space properties, which are much more commonly of interest in molecular simulations. Continue reading