A Theory of Fault-Tolerant Quantum Computation

In order to use quantum error-correcting codes to actually improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a general theory of fault-tolerant operations based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-qubit code. 03.65.Bz,89.80.+h Typeset using REVTEX [email protected] 1 The development of quantum error-correcting codes [1–6] has stirred great hopes for conquering errors and decoherence in quantum computers. However, just the existence of codes, even very good codes, is not sufficient. It is also necessary to be able to perform operations on encoded states without a catastrophic spread of existing errors. However, until now, it was only known how to implement a universal set of gates fault-tolerantly on a few codes [7–9]. While a general formalism for quantum error-correcting codes is known, the code stabilizer [4–6], the theory of fault-tolerant operations is in a great deal of disarray. A quantum gate, unlike a classical gate, can cause errors to spread both forwards and backwards through the gate. The goal of fault-tolerant operations is to prevent the spread of errors within a block, which could change a single correctable error into two errors, which is perhaps more than the code could handle. Even if we use codes that correct more than one error, the spread of errors within a block rapidly reduces the code’s tolerance for errors. Therefore, I define a fault-tolerant operation to be one for which a single operational error can only produce one error within a single encoded block. The assumption is that storage errors on different qubits are independent and that gate errors can only affect qubits which interact via that gate. A transversal operation, in which the operation acts independently on each qubit in the block, is a prototypical fault-tolerant operation. For instance, a bitwise controlled NOT operation from one block to another is fault-tolerant, since errors can spread only between corresponding qubits in the two blocks. Unfortunately, most bitwise operations applied to most codes will not map one valid codeword to another. Until now, there has been no general theory of what a bitwise operation will do to the codewords of a given code. I show below that a bitwise operation will transform the stabilizer of a code. If the stabilizer is rearranged, but otherwise left unchanged, the operation will take codewords to codewords. This will give us a few basic operations on various codes with which to start our analysis. In the quest to perform universal quantum computation, we are not limited to unitary 2 operations. We can also perform measurements. In section III, I analyze the behavior of certain states when a measurement is made. This allows us to see what operations we can derive from the basic operations by using ancillas and making partial measurements of the state. Ultimately, this will allow us to perform universal computation on a large set of codes, among them the five-qubit code, a class of distance 2 codes, and the code encoding 3 qubits in 8 qubits. I. ENCODED NOT AND PHASE Before I advance into the full theory of fault-tolerant operations, I will discuss how to perform encoded NOT and Phase gates on any stabilizer code. The behavior of these gates under more general transformations will tell us what those transformations actually do to the encoded states. The stabilizer S of a code is an abelian subgroup of the group G generated by the operations