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‘Quantum Calculator’ Algorithm Tackles Optimization Problems

ENCODING CONTINUOUS VARIABLES
Before considering how continuous variables have been encoded into
the quantum algorithm, it’s helpful to review the basic degrees of
freedom of a single qubit. Usually, a qubit is represented by a point of a
Bloch sphere, as shown in Figure 1.
Pure states of a qubit correspond to points on the surface of the
sphere and can be defined by the angles ɸ and θ. Similarly, mixed states
of a qubit correspond to points lying inside the Bloch sphere (their
radius coordinate, r, is less than the sphere’s radius) and correspond to
the single-qubit configurations where the qubit is entangled with some
other quantum system, such that the qubit’s reduced-density matrix is
no longer a pure state.
Each qubit in the quantum circuit encodes up to three continuous
variables in the parameters of the Bloch sphere: two angles and one
radius. By combining this with single-qubit quantum tomography, a
variational optimization of the circuit parameters allows the extreme
values of multidimensional functions to be found in a purely analog
and fast way. In other words, Multiverse’s algorithm enables calcula-
tions based on “continuous variables,” which can take on an unlimited
number of values between the lowest and highest points of measure-
ment. Discrete variables, in contrast, can take on only a limited number
of values.

Figure 2: Integral exact and Fourier approximation (a) and
relative error between exact and Fourier approximation (b)
(Source: Multiverse Computing)

The Multiverse technique achieves its
effectiveness by combining two methods for
enabling continuous optimization: quantum
state tomography and the encoding of qubits
Figure 1: Bloch sphere of a qubit (Source: Multiverse Computing)
with three continuous variables.
The Multiverse technique achieves its effectiveness by combin-
ing two methods for enabling continuous optimization: quantum APPLICATIONS
state tomography and the encoding of qubits with three continu- Typical applications of the Multiverse algorithm include mathematical
ous variables. The technique thus makes use of the entanglement, series expansion, such as Fourier series and Taylor expansion, Fourier
superpositions and, now, continuous encoding capabilities of quantum analysis, integrals (complicated integrals can be calculated using
computers. Fourier analysis, so that the integrals become easy to implement) and
“Our research shows that we can transform today’s NISQ devices into differential equations.
advanced quantum-based ‘calculators’ that are able to do very complex Consider the example shown in Figure 2. Given a function I(x),
calculations with very few qubits and limited error correction and Figure 2(a) shows the plot of the exact integral (blue) and the plot
[thereby] provide value now,” said Román Orús, co-founder and chief of the integral calculated using the Fourier series. The plot of the corre-
scientific officer at Multiverse. sponding relative error is shown in Figure 2(b).
The Multiverse team tested the algorithm on a simulator before its Therefore, the new quantum algorithm can help in finance, math-
execution on programmable quantum computers. “These simulations ematics and engineering and is predicted to improve as quantum
are at least comparable to the best classical computers today and will computers become faster and more fault-tolerant. ■
only improve as quantum computing performance increases,” Orús
said. Stefano Lovati is a contributing writer for EE Times Europe.

www.eetimes.eu | MARCH 2023