Errors in quantum computers are of two kinds: sudden perturbations to isolated qubits, and slow random drifts of all the qubits. The latter may be reduced, but not eliminated, by means of symmetrization, namely by using many replicas of the computer, and forcing their joint quantum state to be completely symmetric. On the other hand, isolated errors can be corrected by quantum codewords that represent a logical qubit in a redundant way, by several physical qubits. If one of the physical qubits is perturbed, for example if it gets entangled with an unknown environment, there still is enough information encoded in the other physical qubits to restore the logical qubit, and disentangle it from the environment. The recovery procedure may consist of unitary operations, without the need of actually identifying the error.
Errors in quantum computers are of two kinds: sudden perturbations to isolated qubits, and slow random drifts of all the qubits. The latter may be reduced, but not eliminated, by means of symmetrization, namely by using many replicas of the computer, and forcing their joint quantum state to be completely symmetric. On the other hand, isolated errors can be corrected by quantum codewords that represent a logical qubit in a redundant way, by several physical qubits. If one of the physical qubits is perturbed, for example if it gets entangled with an unknown environment, there still is enough information encoded in the other physical qubits to restore the logical qubit, and disentangle it from the environment. The recovery procedure may consist of unitary operations, without the need of actually identifying the error.
Abstract Quantum computing1—the processing of information according to the fundamental laws of physics—offers a means to solve efficiently a small but significant set of classically intractable problems. Quantum computers are based on the controlled manipulation of entangled quantum states, which are extremely sensitive to noise and imprecision; active correction of errors must therefore be implemented without causing loss of coherence. Quantum error-correction theory2, 3, 4, 5, 6, 7, 8, 9 has made great progress in this regard, by predicting error-correcting ‘codeword’ quantum states. But the coding is inefficient and requires many quantum bits10, 11, 12, which results in physically unwieldy fault-tolerant quantum circuits10, 11, 12, 13, 14, 15, 16, 17, 18. Here I report a general technique for circumventing the trade-off between the achieved noise tolerance and the scale-up in computer size that is required to realize the error correction. I adapt the recovery operation (the process by which noise is suppressed through error detection and correction) to simultaneously correct errors and perform a useful measurement that drives the computation. The result is that a quantum computer need be only an order of magnitude larger than the logic device contained within it. For example, the physical scale-up factor10, 11 required to factorize a thousand-digit number is reduced from 1, 500 to 22, while preserving the original tolerated gate error rate (10−5) and memory noise per bit (10−7). The difficulty of realizing a useful quantum computer is therefore significantly reduced.
A mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.
An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of two parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors are caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such process can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogues of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behave in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such states can be corrected by measurements which determine the error while yielding no information about which codeword or superposition of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysis which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: the quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneously many times before the computation is complete.
The algorithms discussed in Chapter 21 rely on having isolated systems to store information. In practical applications this is not possible, and the environment interacts with the systems causing decoherence. Suppose the data is contained in the state |x〉, and the environment is described by |E〉. The initial state of the entire system is described by the tensor product of the states |X〉⊗|E〉, which evolves according to some unitary operation U. The state |x〉 evolves according to the unitary operation U x which describes the algorithm. In classical error correction codes, all that needs to be corrected are bit flips (see Chapter 10). In the quantum case errors such as bit flips, phase changes and rotations complicate the error correction techniques. Since arbitrary errors in an encoding of information cannot be corrected, only certain types of errors are assumed to occur (this was also the case in classical error correction, where an upper bound in the number of bits that could be flipped in a transmission was assumed). The types of errors depend on the implementation. For example, suppose the types of errors (which we assume are distinguishable due to an encoding) are described by the unitary basis E 1,...E n , so that all errors are a linear combination [138] E = al.E 1 + ... + a n E n ,...E † E = I.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
The quantum computing must be the initial mark of the second technological revolution caused by the deepening knowledge of the humankind about the peculiar features of the quantum mechanics. If we want to unlock all the potential of this new technology, a development of a fault-tolerant quantum computing will be needed. In order to reach this stage, it is necessary an implementation of quantum error correcting codes that guarantee an exponential suppression of the errors at each step of the execution of an algorithm. Besides that, we need to learn how to isolate our qubits aiming a suppression of the decoherence effects. Quantum mechanics imposes various constraints to the process of construction of the error correction codes, one of them is the impossibility of using copies as redundancy mechanism, once no-cloning theorem does not allow the creation of an universal copy machine of quantum states. Another issue to be noted within this context, is that the measurement process causes a collapse in the quantum state, thus leading to a corruption of the information encoded in the state. Fortunately, there are some ways to bypass the problems pointed out through the entanglement and the stabilizer formalism, the first one allows us to recover the redundancy through the delocalization of the information in the Hilbert spaces of the parts that share quantum correlations, while the second one provides a process of gathering information about the errors that can occur in the quantum state without measuring it. The surface codes are one of the quantum codes that use the features highlighted to correct the errors, this type of code is one of the main candidates to be experimentally implemented on a large scale and it presents promising results in the preliminary studies.