n-qubit operations on sphere and queueing scaling limits for programmable quantum computer

We derive a general spherical coordinate formula for a quantum state of n-qubit register and study n-qubit operation rules on \((n+1)\)-sphere with the target to help developing a (photon or other technique)-based programmable quantum computer. The newly developed angle-based n-qubit operation rules are simple and efficient, which reduce the complicated quantum multiplication and division operations to simple addition and subtraction operations just like those used in a conventional computer. The rules for n-qubit operations are realized through measurement-based feedback controls and quantum entanglements. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a \((n+1)\)-sphere) for n-qubit quantum computer-based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the \((n+1)\)-sphere as arrival inputs. In the first regime, the qubit number n is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work-conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number n itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin–Whitt regime.