Quantum Error Correction in correlated quantum noise

The superiority of quantum computation over conventional computation relies on the fact that a quantumbit (qubit) register can be in the superposition of a very large number of classical computational states. At the same time, maintaining coherence of this highly superpositional state poses also the main obstacle for the realization of a quantum computer. For a small number of qubits this difficulty can be overcome by simply reducing the coupling to environmental degrees of freedom, as has been demonstrated by several groups for different physical implementations. With an increasing number of qubits, it will, however, become extremely difficult to reach the required coherence in that way [1]. It is therefore common opinion that a scalable implementation of a quantum computer must use some error correction scheme that recovers the quantum state after it has been distorted by external noise. The existence of error correcting schemes for quantum states, which was shown independently by Shor [2] and Steane [3], is a remarkable fact and has been crucial for the development of the field. The key ideas presented in their work rapidly evolved to a beautiful theory of quantum error correcting codes and subsequently to the concept of fault tolerant quantum computation [4]. Quantum error correcting schemes are usually designed for the independent error model, which by definition does not exhibit correlations between noise of different times and locations. From a physical point of view, this requirement is rather annoying, since in general qubits do interact with a common environment which necessarily introduces some amount of correlations in the noise. To be more specific, in many if not all situations the qubits weakly interact with a common thermal bath of extended bosons (photons and/or phonons). The exchange of bosons between qubits will then cause spatial and temporal error correlations that violate the condition of error independence. Indeed, it has been shown [5] that these kind of processes can lead to drastically enlarged or reduced decoherence of certain states. To which extent do such error correlations interfere with quantum error correction? We have analyzed this problem for optimal Calderbank-Shor-Steane (CSS) quantum error-correcting codes of variable length n (number of physical qubits) and size k (number of logical qubits). As physical noise-model, we use a reduced spin-boson model consisting of n spins – describing an nqubit register – coupled to a common bosonic bath [5, 6]. The amount of noise correlations can be controlled by the inter-spin distance r. Within this framework, we study how code states transform during spin-boson interaction and a subsequent error-correcting operation. The distance between the resulting code state and the initial one – in the sequel denoted as residual error ∆ [cf. Eq. (11)] – serves as a measure for the error-correcting performance of the code. Our main finding is that quantum error correction with CSS codes is substantially hampered by the kind of noise-correlations captured in our model. This becomes evident by the fact that for any fixed information rate k/n > 0 the residual error ∆ approaches a finite constant in the limit n → ∞, unless the spin-boson coupling strictly vanishes or r is infinite (cf. Fig. 1). Using a simple scaling argument we conclude that for a wide range of model parameters CSS codes cannot provide the accuracy needed for large scale quantum computations. In contrast to related studies [7], here the spin-boson coupling is treated in a non-perturbative manner, which we find to be indispensable in the large n limit. Substantial progress towards error correction beyond the independent error model has been made in very recent work [8, 9]. We will briefly comment our results in light of this new work at the end of this Letter. An extended discussion will follow in a future publication [10]. We begin with the physical model for the n-qubit register. It is defined by the Hamiltonian