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Suppose I hand you a quantum computer in an unknown state, but running a known program. You know the program and which part of the program is currently being executed. The program tells the computer to mix up the qubits, then measure one qubit, then mix, then measure, mix, measure, mix, measure, and so on indefinitely. Every time a qubit is measured, you are told the measurement outcome. Qubits are not allowed to decohere without being measured. New qubits/information is never introduced.

You don't care about knowing the initial state of the computer, but you want to eventually know the current state of the computer.

As you are told the measurement outcomes, you learn more and more about the current state of the computer. It would be intractably expensive to track this information in practice, but in principle you could. Eventually, you will know basically all the information relevant to predicting the outcomes of future measurements.

What's the name of the task you are performing? Where can I find papers about it? You're gradually inferring the current state (not the original state!) of an isolated quantum system, but without the ability to do anything to the system except watch what gets measured.

I've tried searching things like "state synchronization", "state estimation", "gradual state inference", and so forth but I always get no results or results for different tasks.

Example

To clarify what I mean, here's an example. Suppose the known program is:

while true:
    CNOT q1 onto q2
    X^(1/8) onto q2
    MEASURE q2

The computer's initial state is unknown to you, but pure: $a |00\rangle + b |01\rangle + c |10\rangle + d |11\rangle$ for some $a$, $b$, $c$, and $d$.

The program starts to run. The measurements of q2 after each iteration start to come in:

ON
OFF
ON
OFF
ON
OFF
ON
OFF
ON
OFF

After that 10'th measurement, what state do you think the system is in? Obviously q2 is OFF, but what's the state of q1?

The measurement result is switching every time, which is far more likely if q1 is ON than if it's OFF. So "probably q1 is ON, but a small chance it's off / some superposition of both" is a pretty good guess.

I checked this guess by simulating the situation in Quirk. I post-selected on those measurement results, and checked that in fact most starting states do get closer and closer to q1 being ON as you progress.

It doesn't matter if it starts close to ON:

simulation of inferrence

Or close to Off:

simulation of inference 2

By the end, q1 is very close to the predicted state: ON.

(The green spheres are state indicators. Notice how the post-selected state of q1 gets closer and closer to Z-up, which is ON. The circuit is cut-off after the 5'th measurement, instead of the 10'th, for space reasons.)

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – David Z
    Apr 7 '16 at 23:34
  • $\begingroup$ Austin Fowler probably has papers on related topics. I would take a look at those and references therein. $\endgroup$
    – DanielSank
    Jun 26 '16 at 18:23

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