Abstract
We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits n, the number of encoded qubits k, the code distance d, the accuracy parameter δ that quantifies how well the erasure channel can be reversed, and the locality parameter ℓ that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy δ that is exponentially small in ℓ, which is the case for perturbations of local commuting projector codes, our bound reads kd2D−1≤O(n(log⁡n)2DD−1) for codes on D-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed O(ℓn(D−1)/D) if δ≤O(ℓn−1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance d and the size d~ of a minimal region that can support all approximate logical operators satisfies d~d1D−1≤O(nℓDD−1), where the logical operators are accurate up to O((nδ/d)1/2) in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.
Highlights
In our study of quantum algorithms, we have found persuasive evidence that a quantum computer would have extraordinary power
A quantum computer will inevitably interact with its surroundings, resulting in decoherence and in the decay of the quantum information stored in the device
If the error superoperator has its support on the set E of all Pauli operators of weight up to t, and it is possible to make a measurement that correctly diagnoses whether an error has occurred, it is said that we can detect t errors
Summary
In our study of quantum algorithms, we have found persuasive evidence that a quantum computer would have extraordinary power. Can be measured to diagnose bit flip errors in the other two clusters of three qubits. We can diagnose in which cluster the phase error occurred by measuring the two six-qubit observables. If two bit flips occur in a single cluster of three qubits, the encoded information will be damaged. Since | ̄0 and | ̄1 are eigenstates of X1X2X3 with distinct eigenvalues, the effect of two bit flips in a single cluster is a phase error in the encoded qubit: X1X2X3 : a| ̄0 + b| ̄1 → a| ̄0 − b| ̄1. We will introduce a phase error into the third cluster in our misguided attempt at recovery, so that altogether Z1Z4Z7 will have been applied, which flips the encoded qubit: Z1Z4Z7 : a| ̄0 + b| ̄1 → a| ̄1 + b| ̄0.
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