Noise in quantum systems can be coherent or stochastic (incoherent). The former is more damaging since such noise can accumulate in one direction and grow quadratically in the number of qubits. While standard quantum error correction (QEC) addresses noise actively by measuring syndromes and applying corrections, we develop conditions for a stabilizer code to passively tackle coherent noise. When the noise introduces a coherent Z-rotation by an angle theta on all qubits, our codes remain unaffected and act as a decoherence free subspace (DFS). Given any [[n,k,d]] stabilizer code and even M, we can produce a [[Mn,k,>= d]] code that is a DFS for this noise.
In this talk, we will begin by revisiting the classical result of MacWilliams that relates the weight enumerator of a code to the weight enumerator of the dual. Then, after reviewing the essentials of stabilizer codes, we will discuss conditions for a transversal Z-rotation exp(ithetaZ) to fix the code space of a stabilizer code. Subsequently, we consider the case where we impose that this transversal rotation fixes the code for all theta, and show that this necessitates the presence of a large amount of weight-2 Z-stabilizers. By organizing these suitably, we will develop DFSs for the aforesaid form of coherent noise. If time permits, we will briefly discuss how we analyze the case where we want only theta <= pi/2^l to preserve the code, in order to induce non-trivial logical gates.