Disrupting cognition: on portfolio optimization problems, machines with the lowest quantum volume perform best
Agnostiq, a quantum computing software company in Toronto, Canada, has released a study that provides the current state-of-the-art for solving combinatorial optimization problems on real, noisy gate-based quantum computers. While this research primarily focuses on the optimization of discrete financial portfolios, it also has broad implications for other important industry issues, including vehicle routing, task scheduling, and facility location services. Related papers can be viewed on the arXiv platform [1].
Running the Quantum Approximate Optimization Algorithm (QAOA) on superconducting (5 IBM quantum computers and Rigetti Aspen-10) and ion trap (IonQ 11-Q) quantum computers to solve the portfolio optimization problem results in quantum volume ( QV) the smallest ibmq_lima performs best. In the past, QV was a key metric for measuring the performance of quantum computers, but new research shows that the QV metric may not be comprehensive.
Defects in quantum volume
Judging by quantum volume benchmarks, it is clear that the overall quality of qubits is improving. Given this improvement in generic metrics, does application-specific performance stay the same? While one might naively think that an improvement in the former means an improvement in the latter, a recent paper in Nature [2] shows that general performance metrics such as quantum volume overestimate what structured programs such as quantum optimization algorithms actually achieve performance. In this article, the researchers explicitly demonstrate improved performance for discrete combinatorial optimization and further expose inconsistencies in general-purpose and application-specific performance.
There are two mainstream approaches to building quantum computers. One is to use trapped ion qubits. Here, the two-level system of a single qubit is realized by the natural electronic states of ions trapped/aligned in an electromagnetic field. The qubits are manipulated by a laser that strikes the ions (ie, their gates are realized by the laser). Another major approach is to use superconducting transmon qubits. Here, an "artificial atom" is created by carefully pairing capacitors and Josephson junctions connected by superconducting wires. In this case, qubits can be manipulated using microwave resonators.
While each mode has its pros and cons, the only way to know which systems perform well in a particular application is to test them. This paper achieves this by performing a combinatorial optimization task on IBM and Rigetti's superconducting transmon QPU and IonQ's trapped ion QPU.
Figure 1 Fidelity of seven quantum computers
How to judge the portfolio optimization performance of each QPU? Given a set of well-known stocks in the tech space (Google, Amazon, FB, Nvidia, Tesla, etc.), each QPU is required to make the most out of the given (i) historical market data and (ii) a fixed number of stocks. The best stock investment option. In order to understand how to judge the performance of each QPU, it is necessary to first understand a property of the underlying algorithm that performs the optimization: the quantum approximate optimization algorithm (QAOA).
In a nutshell, QAOA samples answers to combinatorial optimization problems from a distribution that favors well-sampled solutions. Our metric—Normalized and Complementary Wasserstein Distance (NCWD)—measures how far the QPU deviates from the distribution. Importantly, it is known that adding one parameter (the number of layers p in the QAOA circuit) can improve the bias imposed by the QAOA. Paradoxically, more one- and two-qubit gates are needed to increase p.
That is, since current QPUs are susceptible to noise, there is a point of diminishing returns where any possible improvement in performance is dampened by noise that reduces performance as p increases. Now, a natural question is: at what point do these effects balance out for the QPUs benchmarked in this article?
For regular QAOA (R-QAOA; the original formulation of the algorithm), on IonQ's 11-qubit trapped ion chip, 3 stocks showed a large performance peak at p = 4 [Fig. 2(a)]. For 2 stocks, the performance of the superconducting chip is observed to increase with increasing p until p = 5 [as shown by Rigetti in Fig. 2(b)]. While no fully comparable benchmarks have been performed before, this amounts to a sizable improvement over the most relevant benchmarks in recent years. In fact, just two short years ago [3], it was a milestone to initially observe an improvement in the performance of p; using the performance of two qubits (here equivalent to two stocks) when investigating different optimization problems Increased from p=1 to 2.
Figure 2. Performance of the portfolio optimization algorithm (measured by normalized and complementary Wasserstein distance; NCWD) as a function of the number p of QAOA layers. The qubit topology/connectivity of each QPU is shown as an inset. A performance score of 0.5 means the algorithm is indistinguishable from random guessing, while a score of 1 means the best possible portfolio is sampled with 100% probability. (a) R-QAOA on an 11-qubit IonQ machine. On a larger scale, the performance peaks for 3 stocks at p=4. (b) On Rigetti Aspen-10, two variants of E-QAOA using three stocks and R-QAOA using two stocks. Two variants of three stocks of E-QAOA and two stocks of R-QAOA were used in the wider pool of Rigetti Aspen-10. By p = 5, E-QAOA-II requires more than 1000 one- and two-qubit gates to convert and still produces a good portfolio that is better than random guessing.
This paper also deploys two more advanced versions of QAOA (Quantum Alternation Operator Ansätz) for QPU. These benchmarks have not been performed on quantum hardware for any combinatorial optimization problems, making these measurements a first-of-its-kind. The extension to R-QAOA employs an alternative approach to ensure that the algorithm fits within the budget for the number of shares that can be purchased. In this case, portfolios that are over or under budget are strictly prohibited (at least on an ideal QPU, noise does violate this rule slightly), while R-QAOA allows these portfolios with a penalty to prevent Solutions that violate budget constraints. Here, these variants are called E-QAOA-I (Extended QAOA) and E-QAOA-II.
Each variant takes a slightly different approach, which is fully explained in this article. For both variations, the performance improvement of p is observed [Figure 2(b) shows the performance of 3 stocks with p = 1 to 2 in E-QAOA-I]. Remarkably, the E-QAOA variant yields better portfolios than random guessing, even though more than 1000 one- and two-qubit gates are required. On the Rigetti Aspen-10 QPU, this can be attributed to the native implementation of the XY gates on which the E-QAOA variant relies.
Some surprising observations can be drawn from the R-QAOA benchmark on an IBM quantum computer. IBM provided quantum volume data for these machines, so they could be used to reliably determine whether quantum volume and application-specific performance were consistent—and it turned out they didn't. In fact, the machine with the lowest quantum volume (ibmq_lima, with a quantum volume of 8) actually performs the best in portfolio optimization within the IBM QPU!
Also, performance varies widely across all machines (not just IBM). Fluctuations in portfolio quality of up to 29% were observed in the same measurement. This makes it clear that in addition to any benchmarks, variability must also be measured.
Hardware (and algorithms) are improving dramatically. However, given the researchers’ observations that general-purpose qubit quality benchmarks such as quantum volume are inconsistent with those of 0 in this paper, it’s clear that the best way to track progress is to benchmark machines for specific applications.
Link:
[1] https://arxiv.org/abs/2202.06782
[2] https://www.nature.com/articles/s41567-021-01409-7
[3] https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.14.034010
[4] https://agnostiq.ai/blog/optimizing-financial-portfolios-on-superconducting-and-trapped-ion-quantum-computers/
