Is "probabilitistic,universal, fault tolerant quantum computation" possible with continuous values?

Question

It seems to be a widely held belief within the scientific community that it is possible to do "universal, fault-tolerant" quantum computation using optical means by following what is called "linear optical quantum computing (LOQC)" pioneered by KLM (Knill, Laflamme, Milburn). However, LOQC uses only modes of light that contain either zero or one photon, not more.

Continuous modes of light contain, by definition, much more than one photon. The paper Probabilistic Fault-Tolerant Universal Quantum Computation and Sampling Problems in Continuous Variables Douce et al. (2018) [quant-ph arXiv:1806.06618v1] claims "probabilistic universal fault-tolerant" quantum computation can also be done using continuous modes of squeezed light. The paper goes even further and claims it is possible to demonstrate quantum supremacy using continuous modes. In fact, the paper's abstract says:

Furthermore, we show that this model can be adapted to yield sampling problems that cannot be simulated efficiently with a classical computer, unless the polynomial hierarchy collapses.

A quantum computing startup called Xanadu that has some credibility because it has written several papers with Seth Lloyd, seems to be claiming that they too will ultimately be able to do quantum computation with continuous modes of light, and perform some tasks better than a classical computer.

And yet, what they are doing seems to me to be analog computing (is fault tolerant error correction possible for analog computing?). Also, they use squeezing and displacement operations. Such operations do not conserve energy (squeezing or displacing a mode can change its energy), so such operations seem to require exchanges of macroscopic amounts (not quantized amounts) of energy with an external environment, which probably can introduce a lot of noise into the qc. Furthermore, squeezing has only been achieved in the lab for limited small values, and a claim of universality might require arbitrary large squeezing as a resource.

So, my question is, are these people being too optimistic or not? What kind of computing can be done realistically in the lab with continuous modes of light?

Answer 1

To start off, I would really suggest you to read this review on "Quantum information with continuous variables(cv)". It covers most of your questions with cv architecture. Since it is a very big review, I will try to address your questions with what I can remember from reading that paper and glancing over it again now.

For discrete variables(dv), as you have mentioned, Knill and Laflamme have pioneered the LOQC. But this approach was translated to cvs shortly after the proposal for realization of cv teleportation by Braunstein et al. They showed that cv quantum error correction codes can be implemented using only linear optics and resources of squeezed light.

Now coming to the universality of this type of quantum computer, they have also shown in the paper that a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using linear optics, squeezers and at least one further non-linear optical element such as the Kerr effect(pg.48~50).

I will try to summarize their proof verbally as simple as I can.

1) It is true that, for universal qcs, logical operations can only affect few variables in the form of qubit logic gates and by stacking those gates, it can effect any unitary transformation over a finite number of those variables to any desired degree of precision.

2) The argument is that since an arbitrary unitary transformation over even a single cv requires an infinite number of parameters to define, it typically cannot be approximated by any finite number of quantum operations.

3) This problem is tackled by showing that a notion of universal quantum computation over cvs for various subclasses of transformations, such as Hamiltonians (which are polynomial functions of the operators corresponding to the cvs). A set of continuous quantum operations will be termed universal for a particular set of transformations if one can, by a finite number of applications of the operations, approach arbitrarily closely to any transformation in the set.

4) The result is a very lengthy mathematical proof of constructing quadratic Hamiltonians for EM fields.

So to answer your question, even though, as you mentioned, the squeezing of light adds external noise to qc, I believe that it can be used for error correcting the same noise. Along with that, the claim of quantum speedup arrives from the fact that to generate all unitary transformations given by an arbitrary polynomial Hermitian Hamiltonian (as is necessary to perform universal cv quantum computation), one must include a gate described by a Hamiltonian other than an inhomogeneous quadratic in the canonical operators.

These nonlinear transformations can be used in cv algorithms and may provide a significant speedup over any classical process.

So to conclude, yes cv quantum computation looks optimistic because most of it is theoretical at this point. There are only a few experimental confirmations of the cv architecture like "squeezed-state EPR entanglement", "coherent state quantum teleportation" etc. But the recent experiments in "quantum key distribution" and "quantum memory effect" shows that continuous variable quantum computers have the potential to be as effective as their discrete counterparts if not more for some tasks.