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 , and the states are evolved step by step in time by the application of unitary operators under which a pure state evolves according to(note that unitary operators can entangle states). Thus, the time evolution from a starting time, say , to some time would be given bywhere for each step, the unitary operator is represented by a tensor product of local unitary gates where the index specifies the lattice integer which connects a pair of qubits, and 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 is the circuit's width, and we define its depth as the number of layers of unitary gates. Hence, for the configuration in Figure 1, and . Another way to interpret the circuit is to look at it as a tensor network in which each purple box is a local gate operating on two qubits and the total contraction of qubits indices at the start and the end at time on the lattice integers would give the full unitary time evolution . Thus, the propagation amplitude from some initial state given by the indices to a final state with the indices isOn 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.
Measurements in quantum mechanics are stochastic by nature, which means that circuits with the same exact structure (qubits and gates) would give different outcomes on different runs, see Figure 2. Though this stochastic nature, should be differentiated from randomness. Let be the outcome set of some random measurement, then different measurements on a fixed set of unitary gates would yield distinct records. See the schematic diagram in Figure 2, which sketches a tree diagram with each branch representing a possible outcome of the measurements shown on the circuit. Notice that each measurement results in a different , which would be kind of like a random walk. If our system is just a single qubit, then each measurement causes a jump on the Bloch sphere. However, in the many-body case, the situation is complicated due to correlations between different qubits.
Applications
Near-term quantum computers validation
As we are currently in the Noisy Intermediate-Scale Quantum (NISQ) era, which means that our current quantum computers are not fault tolerance and are not large enough to reach supremacy, we are looking for tasks that have two features: