Decoherence in quantum computing via scattering theory and PDEs
Theoretical quantum computing involves models of computation that obey principles of quantum mechanics. By these principles,
the classical bits of information (usually represented by 0 and 1) are replaced, in an appropriate sense, by coherent superpositions
of quantum states. Regardless of the specifics of realization of a quantum computer,
a building block of any quantum computer is the quantum memory (register). The content of a quantum register is, ideally,
a pure quantum state.
Mathematical models of quantum computing, and also of the quantum memory, usually invoke Hamiltonians that are only
time-dependent, thus leaving out the Laplacian operator - which amounts to omission of the spatial variables.
In a recent formulation by T. T. Wu
the spatial variables
are included and memory operations are formulated through scattering from
the memory (say, of an incident wave of particles or photons). While the reading of the content of the
classical memory is precise, the reading of a quantum memory can be intrinsically erroneous. A common form of error
involves the deviation of the content of the memory from that of a pure state; this kind of error is termed ``decoherence''.
A question that arose from discussions with T. T. Wu and John Myers at Harvard is: What is the decoherence
caused by finite pulses interrogating the simplest non-trivial, two-state, quantum memory? By neglecting decoherence
due to environmental effects, a wave incident on the memory should be of infinite extent for zero decoherence; yet, in
a quantum computer, pulses controlling the content of the memory must be finite and thus
cause decoherence due to their time limits. We modeled the quantum memory via two-channel potential scattering in (1+1)
dimensions by use of the time-dependent Schrödinger equation, and expressed quantitatively how the finite
pulse length causes deviation of the memory content from a pure state.
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1. D. Margetis and J. M. Myers,
Operation-induced decoherence by nonrelativistic scattering from a
Journal of Physics A: Mathematical and General,
Vol. 39, pp. 11567-11581 (2006).
2. D. Margetis and M. G. Grillakis,
quaternions in nonrelativistic scattering from
a quantum memory(PDF),
Journal of Physics A: Mathematical
and Theoretical, Vol. 41, art. 065307, pp. 1-15 (2008).