Quantum Portfolios of Observables and the Risk Neutral Valuation Model

Quantum Portfolios of Observables and the Risk Neutral Valuation Model Fredrick Michael* written Nov 13 2003 August 31, 2021 Abstract Quantum Portfolios of quantum algorithms encoded on qbits have re- cently been reported. In this paper a discussion of the continuous variables version of quantum portfolios is presented. A risk neutral valuation model for options dependent on the measured values of the observables, analo- gous to the traditional Black-Scholes valuation model, is obtained from the underlying stochastic equations. The quantum algorithms are here encoded on simple harmonic oscillator (SHO) states, and a Fokker-Planck equation for the Glauber P-representation is obtained as a starting point for the analysis. A discussion of the observation of the polarization of a portfolio of qbits is also obtained and the resultant Fokker-Planck equa- tion is used to obtain the risk neutral valuation of the qbit polarization portfolio. 1 Introduction Recently, quantum portfolios of quantum algorithms encoded on two-level sys- tems (qbits) have been reported [1]. It has been shown that maximizing the efficiency of quantum calculations can be accomplished via the formation of portfolios of the algorithms and by minimizing both the running time and its uncertainty. Furthermore, these portfolios have been shown to outperform single algorithms when applied to certain types of algorithms with variable running time such as the ”Las Vegas” algorithms and NP-complete problems such as combinatorial search algorithms. In this paper a discussion of quantum portfo- lios of algorithms encoded on qbits and their continuous variables (CV) version (here,harmonic oscillators) is presented. Continuous variables quantum compu- tation performed with harmonic oscillators is discussed in [2, 3, 4]. A quantum portfolio of algorithms is formed and a risk neutral model, analogous to the traditional Black-Scholes and Merton [5, 6, 7] options valuation model, is ob- tained from the underlying stochastic equations. The quantum algorithms are here encoded on simple harmonic oscillator (SHO) states, and a Fokker-Planck equation for the Glauber P-representation is obtained as a starting point for 1 arXiv:1004.0844v1 [q-fin.GN] 2 Apr 2010 the analysis. A portfolio of qbits is also formed and the resultant Fokker-Planck equation of the qbit polarization is used to obtain the risk neutral valuation of the portfolio and measurement option. The results should prove useful in quantum computation and decoherence. 2 Preliminaries A simplified model for quantum computation is proposed wherein the algorithms are encoded on simple harmonic oscillator basis states and are in the presence of a thermal bath, the temperature decoherence effects needing to be taken into account as is usually the case in real-world applications. The quantum computation utilizing harmonic oscillator basis sets representations has been discussed elsewhere [2, 3]. Representations of computation operators or effective operators can be made and the basis sets of these operators can be expressed as harmonic oscillator basis sets. For this case, we will also consider damping, this could be from other noise sources, external or system specific. The harmonic oscillator can be described by a master equation such as (1) Here, ˆa and ˆa† are destruction and creation operators as usual, and ˆρ is the density matrix operator. The mean thermal excitation due to the thermal bath is n = (e ¯hω kBT −1)−1, and Γ is the damping rate [8]. To obtain a Fokker-Planck equation for the SHO, we first suppose that ˆρ has a Glauber P-respresentation [8] (2) Substituting Eq.(2) into Eq.(1) we obtain the Fokker-Planck equation with mixed derivatives (3) The Fokker-Planck equation is put into a regular form if we change variables using quadratures α = x + iy, and Eq.(3) becomes (4) 2 3 Risk Neutral Valuation The Fokker-Planck equation for the SHO implies that there is an underlying stochastic differential equation(s) of the Ito form [8] (5) (6) Here, the Wiener processes dW (noise) are taken to be a Gaussian white noise, and are delta correlated. These stochastic processes can now be included in a portfolio analysis. To construct a portfolio, we first look at the case where the algorithm is encoded on two harmonic oscillators. It can be seen that a generalization to the multi-asset case proceeds straightforwardly. We write the Legendre transform for the N-asset portfolio Π (7) and here, with N = 2 , the function f(x1, x2, y1, y2, t) is the analogue to the (call) option in finance. In our case, it can represent the probability distribution for measurement of the observables of the quantum algorithm(s). These observables are Legendre transformed such that the state function, the portfolio Π , evolves at the known or wanted rate r such that dΠ = rΠdt. The N algorithms evolve independently, regardless of initial phase, and will have differing instantaneous values in their stochastic sample paths. They all have in commo

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