Quantum random circuits (QRC) is a concept of incorporating an element of randomness into the local unitary operations and measurements of a quantum circuit. The idea is similar to that of random matrix theory which is to use the QRC to obtain almost exact results of non-integrable, hard-to-solve problems by averaging over an ensemble of outcomes. This incorporation of randomness into the circuits has many possible advantages, some of which are (i) the validation of quantum computers, which is the method that Google used when they claimed quantum supremacy in 2019, and (ii) understanding the universal structure of non-equilibrium and thermalization processes in quantum many-body dynamics.
Quantum Random Circuits The constituents of some general quantum circuits would be qubits, unitary gates, and measurements. The time evolution of the quantum circuits is discrete in time t ∈ Z {\displaystyle t\in \mathbb {Z} } , and the states are evolved step by step in time by the application of unitary operators U t ≡ U ( t ; t − 1 ) {\displaystyle U_{t}\equiv U(t;t-1)} under which a pure state evolves according to | ψ ( t ) ⟩ = U t | ψ ( t − 1 ) ⟩ {\displaystyle |\psi (t)\rangle =U_{t}|\psi (t-1)\rangle } (note that unitary operators can entangle states). Thus, the time evolution from a starting time, say t = 0 {\displaystyle t=0} , to some time t {\displaystyle t} would be given by U ( t ; 0 ) = U t U t − 1 ⋯ U 3 U 2 U 1 {\displaystyle U(t;0)=U_{t}U_{t-1}\cdots U_{3}U_{2}U_{1}} where for each step, the unitary operator is represented by a tensor product of local unitary gates u τ , x {\displaystyle u_{\tau ,x}} where the x {\displaystyle x} index specifies the lattice integer which connects a pair of qubits, and τ {\displaystyle \tau } is the time step.
Figure 1, shows a time-space diagram of a quantum circuit which shows the local interactions at each time step. In the language of quantum information theory, the number of qubits n {\displaystyle n} is the circuit's width, and we define its depth d {\displaystyle d} as the number of layers of unitary gates. Hence, for the configuration in Figure 1, n = 8 {\displaystyle n=8} and d = 4 {\displaystyle d=4} . Another way to interpret the circuit is to look at it as a tensor network in which each purple box is a local gate u τ , x {\displaystyle u_{\tau ,x}} operating on two qubits and the total contraction of qubits indices at the start t = 0 {\displaystyle t=0} and the end at time t {\displaystyle t} on the lattice integers would give the full unitary time evolution U ( t ; 0 ) {\displaystyle U(t;0)} . Thus, the propagation amplitude from some initial state given by the indices { a 1 a 2 ⋯ a L } {\displaystyle \left\{a_{1}a_{2}\cdots a_{L}\right\}} to a final state with the indices { b 1 b 2 ⋯ b L } {\displaystyle \left\{b_{1}b_{2}\cdots b_{L}\right\}} is ⟨ a 1 a 2 ⋯ a L | U ( t ; 0 ) | b 1 b 2 ⋯ b L ⟩ . {\displaystyle \langle a_{1}a_{2}\cdots a_{L}|U(t;0)|b_{1}b_{2}\cdots b_{L}\rangle .} On the other side, measurements would disentangle the qubits. The used measurements are called projective measurements, defined as observations that leave the degrees of freedom in an eigenstate of the measured operator unchanged.
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