Light quantum computers have made significant progress in solving real-world problems
The current mainstream optical quantum computer is essentially a non-universal quantum device called a boson sampler. The latest Bose sampling experiment of "Nine Chapter 2" is billions of billions of times faster than supercomputers, but there is still a question that needs to be answered: Can the boson sampler be used as a near-term quantum device and enable the so-called "quantum "Computational superiority" becomes useful?
The answer is yes! As a stand-alone device that only produces measurement samples, this seems unlikely, but as part of a quantum-classical hybrid system equipped with a classical optimizer for active feedback and evaluation of objective functions, optical quantum computers can also be used to solve real-world problems. Almost all non-photonic quantum computing platforms have proposed this kind of general-purpose near-term equipment called the variable quantum intrinsic solver (VQE) without active error correction.
According to the recently published ArXiv paper [1], ORCA Computing, an optical quantum computing company headquartered in London, UK, provides a solution.
In view of the limitations of the boson entanglement generated in the linear circuit and the subsequent Fock measurement, they proposed a method of mapping the measurement in the Hilbert space of the boson. Researchers use it to solve examples of the Ising model, which is a purely classical model of particle interaction in statistical physics. In the optimization circuit, this model is called the quadratic unconstrained binary optimization (QUBO) problem. From a practical point of view, QUBO can be used to solve the problem of portfolio optimization.
Variational Boson Solver
In order to solve the problem of insufficient error correction and fault tolerance (this is a common problem of all current quantum computers), the team demonstrated a lighter boson circuit. The entire boson device (called the variational boson solver) is shown in the figure below.
Variational Boson Solver
The quantum state is generated by an m-mode optical interferometer, and its amplitude is a function of the beam splitter and phase shifter angles ϑi and ψi. The output state is measured on the Fock base, and its output is the detection mode-a string of non-negative integers n = (n1,..., NM). The detection mode is mapped to the binary M-tuple b(j) = (b(j)1,..., B(j)M) through the parity function ℘j. Or, directly measure the parity of each mode. Repeat the measurement Ns times to reconstruct the probability of the most probable state. Calculate an objective function that corresponds to the sum of the energy of each bit string (their probability weighting). Then, the objective function is minimized by the classical optimizer, which provides an update of the interferometer parameters ϑi, ψi, and the cycle repeats until convergence is reached.
The parity mapping of n = M℘0
How to solve real problems?
QUBO is usually expressed as the following secondary planning:
Among them, Q ∈ ℝM×M can be written as a symmetric matrix, and x is an M-tuple, such that xi ∈ (0, 1). Finding the optimal solution of the QUBO problem is equivalent to minimizing the classical Ising Hamiltonian. The figure below shows an example of the QUBO problem for a random symmetric matrix Q with dimension M=30.
The energy of the minimum value found is Emin = 41.43
The team used the variational boson solver to achieve C1(70,n) (the shallowest circuit with 70 modes, n = 69, and 70 input photons) sampling. 4 learning curves, corresponding to two parity functions (green and red) with n = 69 photons and two parity functions (blue and orange) with 70 photons, the number of samples Ns = 150. The lowest energy state found by the solver is E = 39.6.
Use the variational boson solver to find the lowest energy
In addition to the QUBO problem, the variational boson solver in this article can also handle other types of optimization problems. Investment portfolio optimization is a very important issue in financial risk assessment. For a given risk, investors need to find investment spreads between N fixed assets to maximize returns. The key idea is that the standard deviation of a portfolio consisting of N assets, that is, risk, is not the sum of N standard deviations of each asset. This means that diversification by considering the correlation between assets can reduce portfolio risk.
Consider N assets with a return of µi (1 ≤ i ≤ N). Denote ∑ as the N-dimensional covariance matrix between these asset returns. If a person invests part of his total investment ωi in asset i, the return µp of the portfolio containing all these assets is:
The risks are:
Where ω = {ωi}1≤i≤N represents the proportion of total investment in each asset. ω ∈ [0, 1]. In the case of a static portfolio, the goal is to minimize the following function:
Additional constraint ∑iωi=1. The parameter γ is the investor's degree of risk aversion. When ω is continuous, the above equation can be used to find the optimal investment. However, when ω is discrete, this problem is difficult to solve.
The team used the variational boson solver to find the lowest risk investment portfolio through the algorithm shown in the table below.
Binary portfolio optimization algorithm.
For the QUBO problem, the equation f(ω) needs to be converted to:
In this equation, B is a constant, much larger than any parameter in the problem. This additional term penalizes the term that does not require the condition ∑iωi=1. This method means that the solver will explore a space with a dimension of 2N, and the effective solution space is a smaller bit string subspace that satisfies ∑iωi=1. For N=20 companies, their weights are encoded as Nq=3 bits, and the solver will explore a space with a dimension of 260 (approximately equal to 1018), but the dimension of the subspace that satisfies the weight configuration of ∑iωi=1 is only 3168. When γ=1, the above equation is solved as QUBO, and the minimum value E=−0.2109 is obtained, and its Sharpe ratio is equal to 4.51.
The learning curve of this method is shown in the figure below. The first 30 iterations have very high energy, which shows how much the solver struggles in the subspace of the bit string that satisfies the constraint.
QUBO: When Nq = 3 and γ = 1, the portfolio of 20 companies is optimized.
In fact, this method is more suitable for non-QUBO optimization problems, such as the following function:
Any ω makes ∑iωi≠0 hold. If ∑iωi = 1, return a value higher than any other value in the question. The advantage of this method is that the algorithm is trained after the candidate solutions are normalized. This means that the dimension of the acceptable solution space is 2N, which is much larger than the way to solve the QUBO problem. Run the same experiment as the QUBO method, and get the minimum Emin=−0.2217, which is equivalent to the Sharpe ratio equal to 6.68. The learning curve of this method is shown in the figure below.
Non-QUBO: When Nq = 3 and γ = 1, the portfolio of 20 companies is optimized.
Performance exceeds classic methods and D-Wave
Richard Murray, CEO and co-founder of ORCA Computing, said: "In this article, we demonstrated that the photonic system can be applied to solve general problems, such as QUBO optimization problems. This paves the way for the practical application of photonics. This kind of application can be implemented faster than a large-scale error correction system."
He added: “ORCA compares the variational boson solver with the existing QUBO solver. When our method is applied to the knapsack problem (an NP-complete problem of combinatorial optimization), its performance exceeds A competitive classical method, as well as the D-Wave quantum method."[2]
Contrast with D-Wave and classic methods.
He said that the method developed by ORCA is very suitable for its PT series of optical quantum computing systems-using proprietary technology and optical fibers to solve quantum computing problems in rack-mounted, portable, and room-temperature systems. ORCA is working with major users of quantum computers to apply the system to real-world problems in the fields of ICT, energy, finance, and defense.
link: [1]https://arxiv.org/pdf/2112.09766.pdf
[2]https://thequantuminsider.com/2021/12/22/orca-researchers-say-photonic-approach-could-tackle-tricky-real-world-problems/
