{"id":313600,"date":"2025-08-03T04:12:21","date_gmt":"2025-08-03T04:12:21","guid":{"rendered":"https:\/\/www.europesays.com\/uk\/313600\/"},"modified":"2025-08-03T04:12:21","modified_gmt":"2025-08-03T04:12:21","slug":"introducing-the-kernel-descent-optimizer-for-variational-quantum-algorithms","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/uk\/313600\/","title":{"rendered":"Introducing the kernel descent optimizer for variational quantum algorithms"},"content":{"rendered":"

In this section, we present a detailed experimental comparison of kernel descent with \\(L=1\\) versus gradient descent, as well as kernel descent with \\(L=2\\) versus quantum analytic descent. As explained in Sect. 2.5<\/a>, these pairings are chosen because of their roughly equivalent computational efforts in terms of circuit evaluations per iteration. The experiments are conducted in two main parts. The first part, described in Sects. 3.1<\/a>, 3.2<\/a>, and 3.3<\/a>, is concerned with randomized circuits and observables. This approach allows us to consider a large, statistically significant number of such circuits and observables. The first part is structured as follows:<\/p>\n

The algorithms will be compared with respect to the quality of the local approximations (Sect. 3.2<\/a>) and their performance in terms of their ability to minimize the objective function (Sect. 3.3<\/a>). Before that, we describe the experimental setup in Sect. 3.1<\/a>.<\/p>\n

In the second part, which is comprised of Sect. 3.4<\/a>, we deal with a problem from the literature on VQAs.<\/p>\n

Setup for experiments with randomized circuits and observables<\/p>\n

Here we describe the setup for our experiments in Sects. 3.2<\/a> and 3.3<\/a>. In these experiments we neither take measurement shot noise nor quantum hardware noise into account. That is, all compared algorithms are provided with exact values of the objective function f computed via statevector simulation.<\/p>\n

In order to ensure independence of our results from any particular choice of circuit or configuration, we repeatedly sample circuits, observables, and points in parameter space randomly in all our experiments. We proceed similarly to the experimental setup in15<\/a>: Points in parameter space are randomly sampled from the uniform distribution on \\([-\\pi , \\pi )^m\\). With the notation from Sect. 2.1<\/a>, the observable \\(\\mathcal {M}\\) is an n-qubit Pauli randomly sampled from the uniform distribution on \\(\\{I,X,Y,Z\\}^{\\otimes n}\\setminus \\{I^{\\otimes n}\\}\\). The generators \\(G_1, \\dots , G_m\\) of the parametrized gates \\(R_1, \\dots , R_m\\) are sampled (independently) from the uniform distribution on \\(\\{I,X,Y,Z\\}^{\\otimes n}\\setminus \\{I^{\\otimes n}\\}\\), and the n-qubit unitaries \\(C_1, \\dots , C_{m+1}\\) are randomly sampled (independently) via the same procedure by which the individual layers in the well-known quantum volume test27<\/a> are sampled. For a visual representation of the entire circuit we refer to Fig. 1<\/a>, the specific choice of the unitaries \\(C_j\\) is explained and visualized in Fig. 2<\/a>.<\/p>\n

Figure 2<\/b>\"figure<\/a><\/p>\n

This figure shows the specific choice of unitaries \\(C_j\\) featuring in the experiments in Sects. 3.2<\/a> and 3.3<\/a>: First, a permutation \\(\\tau _j\\in S_n\\) is sampled uniformly at random. Subsequently, unitaries \\(U^{(j)}_1,\\dots , U^{(j)}_{\\left\\lfloor {n\/2}\\right\\rfloor }\\) are sampled independently from the Haar measure on SU(4). These are then applied to the qubit pairs \\((\\tau _j (1),\\tau _j (2)),\\dots ,(\\tau _j (2{\\left\\lfloor {n\/2}\\right\\rfloor }-1),\\tau _j (2{\\left\\lfloor {n\/2}\\right\\rfloor }))\\) respectively. Note that this is precisely how the individual layers in the quantum volume test27<\/a> are sampled.<\/p>\n

Quality of the local approximations<\/p>\n

Let \\(p\\in \\mathbb {R}^m\\) be a point in parameter space, let \\(g:\\mathbb {R}^m\\rightarrow \\mathbb {R}\\) be a local approximation of f about p. In this subsection we will provide comparisons of the two algorithm pairs kernel descent with \\(L=1\\) versus gradient descent and kernel descent with \\(L=2\\) versus quantum analytic descent with respect to three notions of error at a point \\(\\theta \\in \\mathbb {R}^m\\) that is \u201csufficiently close\u201d to the development point p:<\/p>\n