Algorithmic cooling with N qubits

Quantum computers offer massive advantages over classical computers in terms of execution time and memory efficiency for a subset of problems. While a variety of physical implementations of quantum computers are still being explored, all must fulfill a fundamental set of requirements, one of which is the ability to initialize the qubits into a pure, fiducial quantum state. Such purified, or cooled, qubits are needed for both the initialization of qubits for the computation, as well as for ancillary qubits required throughout the computation for quantum error correction. Thus, a key hurdle in the success of quantum computers is developing methods to generate extremely cold qubits. Algorithmic cooling, a promising technique for purifying qubits, lowers the temperature of a subset of qubits by applying certain logic gates to the entire system of qubits. Here we investigate both optimal and practical algorithmic cooling of a single target qubit using N total qubits in both closed and open systems. We single out the family of algorithms that achieves the minimum temperature for the target qubit in a closed system and find its analytical expression. Within this family, we single out the sub-family of algorithms that achieves maximal cooling with minimal work cost. We also show how changing from a closed to an open system allows for further cooling of the target qubit. When implemented on real devices, distinct cooling algorithms will display distinct performance depending on their complexity. We illustrate the tradeoff between cooling, work cost, and complexity when implementing various N-qubit algorithmic cooling methods on noisy quantum computers. Exploring methods for cooling qubits below levels that physical cooling techniques can practically achieve will bring us one step closer to successful computation with quantum computers in both the near term and the future.