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Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble - and e.g. Ising model is a basic condensed matter model, which is assumed to use Boltzmann ensemble of e.g. sequences of spins - in spatial direction instead of temporal in QM.

Such spatial realization of Wick-rotated quantum mechanics seems to allow to violate Bell-like inequalities, so a natural next question is if we could build Wick-rotated quantum computers in Ising-like systems? For example to be "printed" on a surface, solving encoded problem if assuming Boltzmann ensemble among sequences?

Notice that Wick-rotated QC is different from adiabatic QC - the latter minimizes Hamiltonian, having huge problem with usually exponentially growing number of local minima. The former is closer to Shor - exploits path ensemble, should not have this optimization problem (?)

While quantum computers use unitary gates: with eigenspectrum in complex unitary circle, such Wick-rotated gates would have real eigenspectrum.

Hadamard gate $H$ is used to get initial superposition in quantum computers, below mixing gate $X$ can be used to get (Boltzmann) ensemble in Wick-rotated computers:

$$H=\frac{1}{\sqrt{2}} \left(\begin{array}{cc}1 & 1 \\ 1 & -1 \\ \end{array} \right) \qquad\qquad X= \left(\begin{array}{cc}1 & 1 \\ 1 & 1 \\ \end{array} \right) $$

In theory, controlled e.g. NOT, X should be also possible, the question is what could be realized e.g. in Ising-like system?

While in quantum computers we can only fix initial amplitude in the past, a big advantage of such spatial realization is that we could fix amplitudes in both directions (left and right), what might allow e.g. to solve 3-SAT (NP-complete, end of this arxiv).

In quantum subroutine of Shor's algorithm below, we prepare ensemble (past direction) of all inputs, calculate classical function and measure its value (future direction) - restricting the initial ensemble to inputs giving the same value of classical function - period of such restricted ensemble (found with QFT) gives a hint for the factorization problem.

Analogously for Boltzmann path ensemble for 3-SAT below, but in spatial realization we can also fix values from second direction (right) - restriction (in splits) becomes to inputs satisfying all the alternatives:

enter image description here

Which Wick-rotated gates could be realized in Ising-like systems?

Assuming we could build e.g. above 3-SAT setting, would it work? In other words - is Boltzmann sequence ensemble a perfect assumption, or only an approximation?

Is there a literature for Wick-rotated quantum computers, gates?

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  • $\begingroup$ There is e.g. "Ising formulations of many NP problems" paper frontiersin.org/articles/10.3389/fphy.2014.00005/full but I see that, as in adiabatic quantum computers, it is about finding minimum of Hamiltonian. Wick-rotated quantum computing is different - closer e.g. to Shor than adiabatic: assumes Boltzmann path ensemble instead of minimization, for which problems like exponential number of local minima should not appear (?) $\endgroup$ – Jarek Duda Jan 22 at 12:37
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I'm not exactly sure what you're asking, but note that if you just Wick rotate any old Hamiltonian, you're likely to end up with a path integral with negative Boltzmann weights, which won't actually correspond to any (local) physical statistical system, eg. Ising.

The Hamiltonians that do Wick rotate to a path integral with positive Boltzmann weights are called "stoquastic" and finding their ground state energy has its own complexity class, called StoqMA (contained somewhere in QMA and containing MA). This paper describes the complexity in some detail, but I am not expert enough to summarize it.

I found this nice diagram as Fig. 1 in this paper ("On the complexity of stoquastic Hamiltonians" by Ian Kivlichan... I couldn't find an arxiv link).

enter image description here

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  • $\begingroup$ Wick-rotated version is called e.g. "euclidean path integrals" tpi.uni-jena.de/~wipf/lectures/pfad/pfad6.pdf and is practically used to numerically find ground state. It needs normalization to have stochastic operator, nice intuitions gives discretized: MERW ( en.wikipedia.org/wiki/Maximal_Entropy_Random_Walk ) - just uniform path ensemble, with stationary distribution exactly as in QM ground state. Indeed these gates seem a bit weaker than unitary, but advantage of spatial realization is fixing amplitudes from both sides - do you agree with above 3-SAT construction? $\endgroup$ – Jarek Duda Jan 24 at 19:18
  • $\begingroup$ I can't really digest your diagram, but apparently 3-SAT is in StoqMA since StoqMA contains NP, so there is some reduction. $\endgroup$ – Ryan Thorngren Jan 24 at 19:27
  • $\begingroup$ The construction is that from left we prepare superposition of all 2^w possibilities for w variables. From right we start with preparing |1..1> outputs of m alternatives, which become superposition of 3m spins, such that each triple satisfy OR(x,y,z), then some of them undergo negation NOT, and are connected with left by SPLIT gates enforcing the same value of each variable in the alternatives. Assuming Boltzmann distribution of paths, only intersection of both is allowed - solving problem. Ising model is assumed to use Boltzmann sequence ensemble- so can we solve 3-SAT with Ising realization? $\endgroup$ – Jarek Duda Jan 24 at 20:19

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