SIAM J. COMPUT. c  2007 Society for Industrial and Applied Mathematics Vol. 36, No. 6, pp. 1729–1747 PHYSICAL LIMITS OF HEAT-BATH ALGORITHMIC COOLING ∗ LEONARD J. SCHULMAN † , TAL MOR ‡ , AND YOSSI WEINSTEIN ‡ Abstract. Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid-state NMR and ion traps. “Closed-system” cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance and to the high initial temperatures of some systems. “Open- system” mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an n-qubit device that is originally maximally mixed, but is in contact with a heat bath of bias ε ≫ 2 −n , can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: For bias ε ≪ 2 −n no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit. Key words. quantum computation, state preparation, thermodynamics AMS subject classifications. 68W01, 80A99 DOI. 10.1137/050666023 1. Introduction. Quantum computation poses a difficult experimental chal- lenge. Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in proposals as varied as NMR and ion traps [8, 19, 9, 13, 10, 11]. Such “cooling” (also known as “biasing” or “polarizing”) is required both for initia- tion of the computation [2] and in order to supply ancillas for fault tolerance as the computation proceeds. Cooling of quantum systems has long been essential in a variety of experimen- tal contexts unrelated to quantum computation, and is performed by processes that directly cool the system such as laser cooling in ion traps or application of strong magnetic fields in NMR. Spin exchange has also been employed in order to transfer highly cooled states into the desired system from another that is more readily directly cooled [4, 14, 24]. In all these methods, the temperature is limited by the original cooling process. Algorithmic cooling. It is in principle possible, however, to reach even lower temperatures, by application of certain logic gates among the qubits [22]. (Even prior to quantum computation the need for signal amplification in NMR imaging led to the implementation of a basic 3-qubit logic gate [23].) In several quantum computation proposals this kind of improvement in cooling is necessary due to the requirement that a large number of qubits all be, with high probability, simultaneously in their ground states. We distinguish between closed- and open-system algorithmic cooling methods. In the former [22] an initial phase of physical cooling is performed which reduces the ∗ Received by the editors March 9, 2005; accepted for publication (in revised form) October 6, 2006; published electronically March 19, 2007. http://www.siam.org/journals/sicomp/36-6/66602.html † California Institute of Technology, MC 256-80, Pasadena, CA 91125 (schulman@caltech.edu). The work of this author was supported in part by the NSF (PHY-0456720 and CCF-0524828), the ARO (W911NF-05-1-0294), the Mathematical Sciences Research Institute, and the Okawa Founda- tion. ‡ Technion - Israel Institute of Technology, Haifa 32000, Israel (talmo@cs.technion.ac.il, yossiv@cs. technion.ac.il). The work of these authors was supported in part by the Israel Ministry of Defense and by the Institute for Future Defense Research at the Technion. 1729