The difficulty of detecting local order in
random quantum circuits

Random quantum circuits have emerged as playgrounds to explore quantum many-body physics and principles of equilibration in quantum dynamical systems. There are two major kinds of quantum platforms that are being mostly used to understand some fundamentals of quantum mechanics. One of them are the quantum simulators like ultracold atomic systems where typically one implements a Hamiltonian that causes a continuous evolution of a quantum system in time. The other are digital quantum circuits like superconducting qubit architectures where the quantum state evolves in discrete time steps through specific unitary gates and measurement protocols. Random quantum circuits lie somewhere between these two. Although the random circuits inherit the discrete nature of the applied unitary gates and measurements, their randomized application creates ensembles of states that are reminiscent of a thermodynamic ensemble of quantum states at a finite temperature. As a result, random quantum circuits can harbor phases of matter akin to what one might get in a Hamiltonian system.

However the randomness in random quantum circuits poses a difficulty in measuring some of the phases that can be obtained. When a quantum state is not an eigenstate of a measurement operator, the measurement result is random and the state ‘collapses’ to the eigenstate corresponding to that random measurement result. This phenomena as far as we know is a truly random process. The probability of a particular measurement result however can be found using the Born rule of probability. In a real experiment, one would need to measure an identical state multiple times in order to get enough statistics to determine the probability of different outcomes and gain information about the quantum state. In a random quantum circuit, I will show how these features lead to an inability to distinguish between correlated and uncorrelated spins. This in turn would show that it would be very difficult to detect local spin order in a random quantum circuit.

Let’s imagine we have just two qubits in our system. X1, Z1 represent the Pauli operators for the first qubit and X2,Z2 denote the same for the second qubit. Suppose we have a random circuit where we can get two possibilities. One is where at long times, the two qubits are uncorrelated. In this case, first qubit is in an eigenstate of X1 and the second is in an eigenstate of X2. Since the circuit is random, the qubits can be in any of the +-1 eigenstate of X1 and X2 for a given realization of the random circuit. The second possibility is where the two qubits at long times are correlated and form a bell pair. Now the qubits are in a simultaneous eigenstate of X1X2 and Z1Z2. Once again the random nature of the circuit means that a particular realization can get them into any of the +-1 eigenstates of X1X2 and Z1Z2.

Let’s see what happens qualitatively when you do such an experiment. Consider the case where the qubits are uncorrelated. If we measure both qubits in the Z direction, the four possible measurement results are: both up, both down, first up second down and vice versa.  We have equal probability of getting up or down for both qubits since they are originally in some eigenstate of X1 and X2.  Thus when we repeat the circuit and the measurements, we get equal probability of all four outcomes. Now what if we have the case where the qubits are correlated? It turns out the measurement results would be exactly identical to the uncorrelated case. Let’s see how that occurs.

Suppose in one rendition of the random quantum circuit, the correlated qubits are in +1 eigenstate of X1X2 and Z1Z2. In this case, when we measure the qubits in the Z direction, we get that they are either both up or both down. But another rendition of the random circuit might place the system in a +1 eigenstate of X1X2 but -1 eigenstate of Z1Z2. Now the measurement results we would get are: first up second down and vice versa. When we repeat the circuit and the measurements a large number of times, we once again get equal probability of all the four possible measurement results. This is despite the fact that in this case, the qubits are genuinely correlated. The issue of course lies in the fact that the sign of the correlation (+1 vs -1) is random and that makes it look like the uncorrelated case.

That’s the chief issue in measuring a spin ordered phase in a random circuit. We detect ordered phases by measuring a local order parameter or local correlation functions. In random quantum circuits, the order parameter of an ordered phase can fluctuate between positive and negative values. When averaged, it becomes indistinguishable from the unordered phase. Only the square of such an order parameter which gives +1 for both +1 and -1 eigenstates can distinguish correlated and uncorrelated spins. Such squares of order parameters or non-linear functions are easy to calculate in theiry but hard to measure in an experiment. The chief culprit of this issue is the randomness of quantum measurements. One possible but highly inefficient way of mitigating this problem is by post selection. We can force the system to remain in let’s say, the +1 eigenstate of any measurement that we do by rejecting any instances where we get a -1 result. Of course, the probability of getting all +1 results goes exponentially down with increasing number of qubits in our system.

It would be an interesting avenue to find an efficient way to detect order in random quantum circuits.