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The Essence of Quantum Computing

No-Go theorems. Quantum superposition imposes certain severe restrictions on quantum systems, colloquially known as the No-Go theorems. They include the No-Cloning theorem54 (the impossibility of cloning a qubit of unknown state), the No-Deletion theorem (the impossibility of deleting a qubit of unknown state using unitary operators), and the No-Hiding theorem55 (quantum information cannot be completely hidden in correlations56). Quantum mechanics also forbids the transmission of classical information using only quantum mechanics (No-Communication theorem57). Recently, Luo, et al have investigated general quantum transformations forbidden or permitted by the superposition principle for various goals58.

Chaos. Physics admits “deterministic” chaos (e.g., in weather systems, nonlinear pendulums, etc.), a special type of disorder in classical physics, but not in quantum physics. A classical system, in a state of deterministic chaos, never retraces any part of its trajectory in phase space. A quantum system, even if it appears to mimic chaos, say, the probabilities of the wave function evolve in a complicated way for a long period of time, the system will eventually start to repeat itself. Yet it has an enigmatic relationship with classical chaos59, an instance of an enigma wrapped in a mystery60. It turns out that quantum systems do interesting things when their classical counterparts go crazy.

A plausible argument against quantum chaos is Heisenberg’s uncertainty principle since it does not entertain classical trajectories of particles. Therefore, one cannot assess sensitive dependence on initial conditions or exponentially diverging trajectories in quantum mechanics. Furthermore, In the Schrödinger equation,

where the first term on the right-hand side represents the kinetic energy and the second term the potential energy related to some conservative external field V = V(r, t) which is a function of position and time. The ∇2 operator is a ‘smoothing operator’. The value of  ∇2F(x, y, z) is the average of the values F(x, y, z) takes in some neighborhood of (x, y, z). To appreciate this point, recall that the solutions F(x, y, z) to Laplace’s equation ∇2F(x, y, z) = 0 are so analytically smooth that they possess derivatives of every order.

A reader new to quantum chaos can start with Barry Cipra61. The paper describes conjectural links between the Riemann zeta function related to prime numbers, and chaotic quantum-mechanical systems, and then read Martin Gutzwiller62 who produced a key formula, called a trace formula that showed how classical chaos might relate to quantum “chaos”. Gutzwiller is credited with the first investigation of relations between classical and quantum mechanics in chaotic systems.

Intriguing complexity. Finally, it is intriguing that the state vector |Ψ〉 by postulate 1 is a complex entity with real and imaginary parts. While complex functions do appear in classical physics, they are invariably interpreted either as (1) an indicator that the solution is unphysical, e.g., in the Lorentz transformation when the speed of a physical object is greater than the speed of light, or (2) as a shortcut to dealing with two independent and equally valid solutions of the equations, one labelled real and the other imaginary, as in the case of two dimensional inviscid flow governed by the Laplace equation where the stream function and the velocity potential appear jointly as a complex variable. In quantum mechanics, |Ψ〉 as a whole rather than its real and imaginary parts separately, is of interest. Yet, since |Ψ〉 is not directly observable and is not a real physical entity, its complex character can be ignored, especially in view of postulate 3, which, in any case extracts a real value in a measurement. All measurable quantities depend on the absolute squares of the components of the state vector and such absolute squares are always real. At the quantum level, why does Nature prefer Hilbert space and complex variables? Why is √(-1) so central to Nature? We do not know.

[54] Wootters & Zurek (1982); Peres (2002) (As Asher reports, the title of the paper was contributed by John Wheeler.)
[55] Pati & Braunstein (2000).
[56] Braunstein & Pati (2007); Pati (Undated).
[57] Peres & Terno (2004).
[58] Luo, et al (2017).
[59] Pool (1989).
[60] Churchill (1939). The phrase is an adaptation of Winston Churchill’s remark about Russia during World War II: “It is a riddle wrapped in a mystery inside an enigma”.
[61] Cipra (1999).
[62] Gutzwiller (1990).

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