We Need A New Way to Benchmark Parameterized Quantum Circuits

By Hannah Sim, Graduate Student in Chemical Physics at Harvard

In recent years, there have been significant developments in variational quantum algorithms — algorithms that use both quantum and classical computers to accomplish a particular task. Two early examples of such algorithms are the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA), both of which can be implemented using Qiskit! Since then, scientists have extended the variational algorithm framework for applications in factoring, generative modeling, quantum data compression, quantum circuit compiling, and more.

While it’s exciting to see all of the prospects of near-term quantum computing, a big challenge lies in improving these algorithms to reliably run larger problem instances on today’s small and noisy devices. Scientists are constantly coming up with improvements, ranging from error mitigation techniques to better energy estimation schemes and optimization algorithms. In this blog post, we wanted to investigate another key component of variational quantum algorithms: parameterized quantum circuits, and how to pick the right one for your algorithm. We don’t have the answer for that yet, but we’ve proposed two new concepts, called expressibility and entangling capability, that will help us make that decision.

What are parameterized quantum circuits?

In many variational algorithms, there is a common “recipe” or framework:

In this recipe, one crucial ingredient is the parameterized quantum circuit (PQC). The parameterized quantum circuit can be thought of as the interface between quantum and classical computers: the quantum computer executes the PQC to estimate some quantity, e.g. energy of a prepared state in VQE, and the classical computer tunes the parameters that came out of the quantum circuit to improve the parameterized state. These new parameters go back into the quantum component, and the cycle continues.

In applications such as quantum chemistry, researchers have proposed and developed effective PQCs such as the unitary coupled-cluster (UCC). The UCC circuit is based on a method called coupled-cluster which is considered the “gold standard” in computational chemistry, and can produce states that can well-approximate the ground states of various types of molecules. However, the UCC circuits, in practice, are too deep — they cannot easily be implemented on today’s quantum computers. To circumvent this problem, researchers at IBM produced the so-called “hardware-efficient” ansatz, a shallow parameterized quantum circuit that is composed of gates that are natural operations on the quantum hardware. Similar circuit designs have also been applied to applications in quantum machine learning.

While various works have proposed different PQC designs, from a variational algorithm user’s perspective, how does one choose which circuit to use for their algorithm? That is, if you’re given a “toolbox” of five different parameterized quantum circuits, which circuit do you apply to your variational algorithm and why?

This is a difficult question to answer, but our work [1] tries to approach this question by first trying to come up with properties (or descriptors) of PQCs that we can use to distinguish among circuit designs and identify circuits that have limited capabilities. Two such descriptors are expressibility and entangling capability.

Expressibility

Expressibility is a measure of a parameterized quantum circuit’s ability to generate states from the Hilbert space. In practice, to quantify expressibility, we compute the extent to which the set of states we generate from a circuit deviates from the (expressive) uniform distribution in the state space. Why the uniform distribution? This is because (1) we know that by sampling the uniform distribution in the state space, we can, in theory, express any state in the Hilbert space. Therefore, it is a suitable reference case to compare our set of states sampled from a PQC. And (2) this uniform distribution has mathematical properties that make computation of the deviation easier.

For instance, consider the following single-qubit circuits:

1. Circuit A consists of an identity gate. This means that the output state is always the |0> state. This circuit is not expressive in terms of the states the circuit can explore.
2. Circuit B has a parameterized RZ gate following a Hadamard gate. By tuning the RZ parameter value, you are exploring the equator of the Bloch sphere.
3. Circuit C is similar to circuit B but has an additional parameterized RX gate. This provides an additional degree-of-freedom to move about the Bloch sphere.
4. Lastly, as reference, we have circuit D, a uniformly sampled unitary U that uniformly maps the |0> state to any point on the Bloch sphere. That is, this circuit is able to express any state on the Bloch sphere with equal probability.

If we were to sample 1000 different output states for each circuit (for parameterized quantum circuits, we uniformly sample the parameter vectors and simulate the corresponding output state), we’d observe something like this:

For circuit A, all 1000 points are located on the “North Pole” of the Bloch sphere (i.e. where |0> is located). For circuit B, we observe output states distributed about the equator of the Bloch sphere. For circuit C, we see that we can cover all of the Bloch sphere with the output states but the coverage is not uniform; there are concentrations of points on the +X and -X poles of the Bloch sphere. Lastly, as expected, we see a uniform coverage of points on the Bloch sphere for circuit D.

Expressibility is a way to quantify what we just observed for these single-qubit circuits, where one can identify circuits with limited expressive powers, such as circuit A. In the example above, circuit A would have a low expressibility score due to the limited set of states it can explore, whereas circuit C and circuit D would have high expressibility scores. While future investigations are needed to determine if expressibility is a “good” property for parameterized quantum circuits, it is one way we can distinguish among circuit designs and identify circuits with limited expressive powers.

Entangling capability

Entanglement is often seen as a key resource in quantum computing. We wanted to come up with a way to quantify how much entanglement a parameterized quantum circuit can generate. To do so, we used the Meyer-Wallach measure [2], a scalable entanglement measure for multi-particle (multi-qubit) systems. Computing the Meyer-Wallach measure is one way you can quantify how entangled a given state is: the Meyer-Wallach measure of a product (i.e. unentangled) state is 0 while the Meyer-Wallach measure of a highly entangled state, such as the Bell state, is 1.

We ran the following analysis:

1. Generate k sample parameter vectors for a parameterized quantum circuit C.
2. Simulate k corresponding parameterized quantum states.
3. Compute the Meyer-Wallach measure for each parameterized quantum state.
4. Compute the average Meyer-Wallach measure value.

With the average Meyer-Wallach measure value for each PQC, we could determine which circuits generate more highly entangled states. For a simple example, consider the two parameterized quantum circuits:

The circuit on the left has no entangling operations (e.g. two-qubit gates), so it has no entangling capability. Thus, this circuit would produce states that have Meyer-Wallach measures of 0, leading to an averaged value of 0. On the other hand, the circuit on the right has several two-qubit gates and thus is able to generate quantum states with some entanglement. Therefore, the average Meyer-Wallach measure would be greater than 0. Such insight may be useful for cases like VQE problems, in which we are often trying to prepare ground states that may be highly entangled. In this scenario, testing the algorithm with a circuit that can readily produce highly entangled states may be advantageous.

Using the two descriptors, expressibility and entangling capability, one can identify circuits that have limited capabilities. We expect that such circuits are likely poor candidates for variational algorithm instances. We additionally computed other descriptors such as the numbers of two-qubit gates and parameters a circuit has in order to determine which circuits are “cheaper” to implement on quantum devices and optimize. With all of this information, we can have a better understanding of which circuits are expressive and able to generate entangled states but are also relatively cheap to implement. We can then select circuits with these properties to test first on variational algorithms. This is the circuit design and selection process we used in our latest work on “quantum computer-aided design” (QCAD) [3], where we designed a PQC based on an existing circuit with favorable descriptor values (i.e. relatively high expressibility and entangling capability with moderate parameter count).

Future work

While our work proposed several descriptors, this is only the first step. In order to more confidently use these descriptors to guide our circuit selection process, we need to verify whether expressibility and/or entangling capability are in fact figures-of-merit. One way to approach this would be to run benchmark studies relating the descriptors to algorithm performance. Fortunately, several works have started research in this direction, investigating the relationship between the descriptors with performances of VQE [4] and the variational quantum classifier [5].

References

1. Sim, Sukin, Peter D. Johnson, and Alán Aspuru‐Guzik. 2019. “Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms.” Advanced Quantum Technologies, October, 1900070. https://doi.org/10.1002/qute.201900070.
2. Meyer, David A., and Nolan R. Wallach. 2002. “Global Entanglement in Multiparticle Systems.” Journal of Mathematical Physics 43 (9): 4273–78. https://doi.org/10.1063/1.1497700.
3. Kyaw, Thi Ha, Tim Menke, Sukin Sim, Nicolas P. D. Sawaya, William D. Oliver, Gian Giacomo Guerreschi, and Alán Aspuru-Guzik. 2020. “Quantum Computer-Aided Design: Digital Quantum Simulation of Quantum Processors,” June. http://arxiv.org/abs/2006.03070.
4. Nakaji, Kouhei, and Naoki Yamamoto. 2020. “Expressibility of the Alternating Layered Ansatz for Quantum Computation.” http://arxiv.org/abs/2005.12537.
5. Hubregtsen, Thomas, Josef Pichlmeier, and Koen Bertels. 2020. “Evaluation of Parameterized Quantum Circuits: On the Design, and the Relation between Classification Accuracy, Expressibility and Entangling Capability.” http://arxiv.org/abs/2003.09887.