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Quantum mechanics can be described as a Markov chain? That seems plausible but I haven't worked with MCs enough to see exactly how. Could you please elaborate?

Quantum mechanics can be described as a Markov chain? That seems plausible but I haven't worked with MCs enough to see exactly how. Could you please elaborate? It seems interesting.

zeofig 2y ago
A Markov process is a random process where the new state only depends on the old state, not anything else. This can be stretched to include almost anything, since you can expand the definition of the state to record history or whatever you want, although you may make the process much more difficult to work with mathematically. In other words, the fact that something may be a Markov process is generally not interesting, because it is a very broad definition, and doesn't guarantee any interesting properties without further assumptions. The wavefunction in QM doesn't evolve randomly, so I would say it is not technically a Markov process. On the other hand you can create a theory of observables derived from QM that is "random" (depending on your interpretation of quantum mechanics).
- evanb 2y ago
  You will have a hard time constructing Markov Chain that correctly models the real-time evolution of physical quantum-mechanical observables. The problem is that the transition matrix that governs quantum-mechanical evolution is exp[i t H], which is not a well-formed probability distribution.
evanb 2y ago
If you want to study a quantum mechanical system in equilibrium at inverse temperature β, the interesting quantity is the partition function Z = tr exp[-β H]. This can be converted into a path integral Z = ∫ dφ exp[-S[φ]] which can be importance-sampled via the Metropolis-Hastings algorithm [mh] via Markov-chain Monte Carlo. This approach is commonly used in lattice field theory [lft], where the Hamiltonian H is that of a discretized spacetime (or the problem is formulated in terms of the action S to begin with). Real-time problems in quantum mechanics involve exp[i t H] which brings a horrible complication called the sign problem [sign]. The one-sentence summary is that exp[-β H] is positive-definite but exp[i t H] is not and it's not clear how to incorporate a complex Boltzmann weight as a probability for MCMC. mh: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_al... lft: https://en.wikipedia.org/wiki/Lattice_field_theory https://en.wikipedia.org/wiki/Lattice_field_theory sign: https://en.wikipedia.org/wiki/Numerical_sign_problem https://en.wikipedia.org/wiki/Numerical_sign_problem
ben_w 2y ago
I would have said something very close to zeofig's answer, with the one significant change being to say that a deterministic system is itself a very boring kind of Markov chain where all states' transition probabilities are exactly 0 or 1.

Quantum mechanics can be described as a Markov chain? That seems plausible but I haven't worked with MCs enough to see exactly how. Could you please elaborate? It seems interesting.

zeofig 2y ago

A Markov process is a random process where the new state only depends on the old state, not anything else. This can be stretched to include almost anything, since you can expand the definition of the state to record history or whatever you want, although you may make the process much more difficult to work with mathematically. In other words, the fact that something may be a Markov process is generally not interesting, because it is a very broad definition, and doesn't guarantee any interesting properties without further assumptions. The wavefunction in QM doesn't evolve randomly, so I would say it is not technically a Markov process. On the other hand you can create a theory of observables derived from QM that is "random" (depending on your interpretation of quantum mechanics).

evanb 2y ago
You will have a hard time constructing Markov Chain that correctly models the real-time evolution of physical quantum-mechanical observables. The problem is that the transition matrix that governs quantum-mechanical evolution is exp[i t H], which is not a well-formed probability distribution.

evanb 2y ago

If you want to study a quantum mechanical system in equilibrium at inverse temperature β, the interesting quantity is the partition function Z = tr exp[-β H]. This can be converted into a path integral Z = ∫ dφ exp[-S[φ]] which can be importance-sampled via the Metropolis-Hastings algorithm [mh] via Markov-chain Monte Carlo. This approach is commonly used in lattice field theory [lft], where the Hamiltonian H is that of a discretized spacetime (or the problem is formulated in terms of the action S to begin with). Real-time problems in quantum mechanics involve exp[i t H] which brings a horrible complication called the sign problem [sign]. The one-sentence summary is that exp[-β H] is positive-definite but exp[i t H] is not and it's not clear how to incorporate a complex Boltzmann weight as a probability for MCMC. mh: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_al... lft: https://en.wikipedia.org/wiki/Lattice_field_theory https://en.wikipedia.org/wiki/Lattice_field_theory sign: https://en.wikipedia.org/wiki/Numerical_sign_problem https://en.wikipedia.org/wiki/Numerical_sign_problem

ben_w 2y ago

I would have said something very close to zeofig's answer, with the one significant change being to say that a deterministic system is itself a very boring kind of Markov chain where all states' transition probabilities are exactly 0 or 1.