Continuous Symmetries and Approximate Quantum Error Correction

Abstract

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the errorcorrection infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

New perspectives on covariant quantum error correction

“…In particular, any finite-dimensional local-error-correcting quantum code only admits a finite number of transversal logical operations, which forbids the existence of codes covariant with continuous symmetries (discrete symmetries are allowed though [9,10]). More generally, quantum codes under symmetry constraints, namely covariant codes, are of great practical and theoretical interest.…”

Approximate Bacon-Shor code and holography

“…It is known that AdS/CFT corresponds to an exact quantum error correction code only in the limit of N → ∞, where the bulk gravitational interaction controlled by G N goes to zero. For finite N or non-negligible gravitational interaction in the bulk, it has been argued that error correction must be approximate [27][28][29]. Despite these insights, the exact manner in which gravity arises in these tensor network quantum error correction codes remains unexplored.…”

“…Indeed, when A = {1, 2} one has access to the subalgebra generated byX. Now the decomposition for this subalgebra is along basis vectors 29) which are both separable with respect to this bipartition. In particular, with decoding unitary U A = CN OT (1, 2) we have…”

“…This construction is to ensure that we can efficiently contract these approximate tensors when evaluating the correlation function. 29 For the sake of simplicity, we fix the amplitudes of the noise term for the double copy Bacon-Shor tensors to be α i =0 = γ andγ = 2|γ| 2 . Similarly, set β <i,j> = β and the constant factorδ(β) now is the same for operator insertions at all 5 legs.…”

“…By the (approximate) Eastin-Knill theorem [29,90,91], it is known that the (non-trivial) continuous time evolution in AdS/CFT and perfect erasure correction on the boundary appear incompatible [92]. It will be informative to further elaborate on the relationship between AQEC and continuous time evolution using our explicit construction.…”