Quantum Computer Visualization without Wavefunctions - Lemmygrad

The math behind quantum computing looks pretty similar to classical probabilistic computing. The only real difference is that the vector you evolve, ψ, is made up of complex-valued probability amplitudes rather than just being a vector of real-valued probabilities. I have been trying to bring them closer together and make quantum computing look more similar to probabilistic computing. The update rule for probabilistic computing is the following: - p⃗’ = Γp⃗ Where p⃗ is your probability vector, Γ is a stochastic matrix describing the probabilistic logic gate, and p⃗’ is the updated probability vector after the probabilistic logical operation is applied. Now, if you take ψ and break it up into two real-valued vectors such that x⃗=ℜ(ψ) and y⃗=ℑ(ψ) and then transform x⃗ and y⃗ from Cartesian coordinates to polar coordinates, you get two new vectors: p⃗ and φ⃗. Notice how we have p⃗. This very much interested me, so I tried to write down an update rule for this p⃗ directly, without going back to the original ψ formalism, and, interestingly, the rule ends up looking like this. - p⃗’ = Γp⃗ + f(φ⃗) It looks identical to the probabilistic computing update rule plus an additional non-linear term that depends upon φ⃗. I didn’t realize this until later, but apparently pilot wave theory does a similar thing, by putting ψ into polar form it gets a classically evolving probability distribution + an additional non-linear term called the quantum potential. It’s more similar to probabilistic computing, but now I need to give some way to make sense of φ⃗. This problem is easier than making sense of ψ since φ⃗ doesn’t carry information that looks like possible paths as p⃗ does, so, regardless of the interpretation, we have already bypassed the problem of devolving into multiverse woo. I found that a good way to visualize such that φ⃗ is a property of the system is to treat it as a “connective” property. Imagine plotting each bit in a quantum computer as a node on a hypergraph. Inside the node holds the bit’s “intrinsic” property, that being a bit value of either 0 or 1. Then, you draw all the edges on the hypergraph and place numbers on each edge, each ranging from -2π to 2π. These numbers are weighted connections and represent the connective properties of the system. Below is an illustration using 3 qubits. [https://lemmygrad.ml/pictrs/image/1cd0a3a0-0646-4fcb-8eaa-ef0b3e9b5f2c.png] Note that the entire ontic state of the system (the intrinsic and connective properties) are all real-valued. This is the key distinction between probabilistic and quantum computing: quantum computing also has a deterministically evolving set of connective properties between the bits, which I refer to as the phase network, that you have to keep track of, and the state of this network can alter the behavior of logic gates, meaning that the same logic gate could have different stochastic behavior on the bits if it is in a different state. What is particularly interesting is that if you represent it in this form, then there is no longer a “collapse of the wavefunction.” Measurement can just be represented as a Bayesian update on p⃗ since p⃗ is epistemic. Since φ⃗ is ontic, you do not need to touch it when you make a measurement. We thus reduce the “collapse of the wavefunction” to a classical knowledge update just like in probabilistic computing. Of course, we all know that systems behave differently if you measure them vs if you don’t. But in this framework, this becomes explained slightly differently. It is not that, if you measure something, you immediately “collapse” anything. Rather, you can demonstrate that if particle A interacts with particle B such that the interaction is entirely passive on A but has the effect of recording A’s value onto B, then it will influence the future nomology of A. By nomology I mean, the intrinsic properties of A do not immediately change as a result of this, but its behavior in a future interaction can change. I refer to this as the “information sharing principle” (ISP), that even if an interaction is passive, if it shares information about a system’s state, then the stochastic laws can lead to a different future evolution of the system. Hence, in something like the Mach-Zehnder interferometer, the measurement of the which-way information can be said to passively reveal the location of the photon on a particular arm of the interferometer, but by sharing its information with the measuring device, it will alter how it later is affected by the second beam splitter. You thus do not need to explain this “observer effect” with an immediately perturbing interaction, as if you physically “collapsed” something by looking at it. To actually fully illustrate the evolution of the ontic states, however, I would need some method of computing transitional probabilities. Transitional probabilities are a form of conditional probabilities where we condition on the immediate state before. For example, rather than asking the probabilities that the qubits are in a particular state at time t=4, we ask what are the probabilities that the qubits are in a particular state at time t=4 while conditioning on knowledge of the qubit values at time t=3. In traditional quantum mechanics, if I were to ask what are the transitional probabilities, the question doesn’t even make sense, because it denies that systems have real values when you are not looking, and so it has no values at t=3. Of course, if you tried to measure those values at time t=3, you would alter the behavior of the computer due to the ISP, and it wouldn’t be the same algorithm anymore. If quantum computing really is a kind of probabilistic computing, then we should be able to imagine that if we were Laplace’s demon and could observe the qubits at time t=3, that we could then condition on them without altering the future evolution of the system via the ISP and then condition on that to update our probabilities for the state that the system will be in at time t=4. It turns out we can, indeed, do this, and it was first figured out by John Bell who showed that you can fit QFT to a stochastic process. The neat thing about transitional probabilities is once we have a way to compute them, then we can evolve the system stochastically such that at each set of logic gates applied to the qubits, we compute the transitional probabilities, then choose a new configuration of the qubits weighted by those transitional probabilities. The system will then evolve such that it has a definite bit value at every step of the algorithm, but if you ran the algorithm over and over again, each intermediate step would correspond to the intermediate Born rule statistics. I have put this all together with a website. In the website, there is a simulator for a 3-qubit quantum computer and a drag-and-drop interface to place quantum logic gates onto the tracks, and you can press Play to run them. On the left-hand side is the “epistemic state” which is p⃗, the statistical distribution for the bits at that moment in the algorithm. On the right-hand side is the “ontic state,” which is both φ⃗ (the connective properties represented by edges on the hypergarph) and the bit values of the qubits (the intrinsic properties represented by a bit value of 0 or 1 within the node). [https://lemmygrad.ml/pictrs/image/23d4b9c2-944a-4363-9290-1cc9a88ee944.png] As you will notice if you play around with the drag-and-drop interface and make different circuits, you will see that the qubits always possess an ontic state, a measurement is just a Bayesian knowledge update on the epistemic state and does not affect the ontic state in the moment of measurement, you get correct transitional probabilities, The link is at: - https://www.stochasticnetwork.com/ [https://www.stochasticnetwork.com/] Please click the “EXPLANATION” button on the top-right if you want to see a technical explanation of all the equations and such I am using. I do not use ψ at all in the entire code of the simulator. It only evolves it directly using p⃗ and φ⃗. The proper equations for that are in the document you can find in “EXPLANATION” if you want to see them. Note that if you have any questions/concerns it is probably already addressed in that document.