Multi-Error-Correcting Amplitude Damping Codes Runyao Duan $^{∗†}$, Markus Grassl$^{‡}$, Zhengfeng Ji$^{§}$, and Bei Zeng¶ $^{∗}$Centre for Quantum Computation and Intelligent Systems (QCIS), Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW 2007, Australia $^{†}$State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China $^{‡}$Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore $^{§}$Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L2Y5, Canada $^{¶}$Institute for Quantum Computing and the Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L3G1, Canada designing codes for this situation and in studying their fault tolerance properties [1], [7], [8], [15], [23]. All those works deal with error models which are still described by Kraus operators that are Pauli matrices (Pauli Kraus operators), but the X - and Y -errors happen with equal probability p$_{x}$ = p$_{y}$ , which might be different from the probability p$_{z}$ that a Z - error happens. The quantum channels described by this kind of noise are called asymmetric channels E AS acting on any one-qubit quantum state ρ as Abstract —We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure- correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction. Index Terms —Amplitude damping channel, quantum error correction, concatenated quantum codes, quantum erasure code. The choice p$_{x}$ = p$_{y}$ is related to a physically realistic error model including amplitude damping (AD) noise and phase damping noise [22]. The Kraus operators for AD noise with damping rate γ are In most of works on quantum error correction, it is assumed that the errors to be corrected are completely random, with no knowledge other than that they affect different qubits inde- pendently [22], [9]. Or, equivalently, this is to assume that the Pauli-type errors X = ( 0 1 $_{1 0}$) , Y = ( 0 − i i 0 ) , and Z = ( 1 0 0 − 1 ) , happen with equal probability p$_{x}$ = p$_{y}$ = p$_{z}$ = p/ 3 . The quantum channel described by this kind of noise is called depolarizing channel E $_{DP}$. The most general physical operations (or quantum chan- nels) allowed by quantum mechanics are completely positive, trace preserving linear maps which can be represented in the following Kraus decomposition form: Hence the linear span of the operators A$_{1}$ and A † 1 equals the linear span of X and Y . If the system is at finite temperature, the Kraus operator A † 1 will appear in the noise model [22]. Thus, if the code is capable of correcting t X - and t Y -errors, it can also correct t A$_{1}$ - and t A † $_{1}$-errors. where A$_{k}$ are called Kraus operators of the quantum channel N and satisfy the completeness condition ∑ $_{k}$A † $_{k}$A$_{k}$ = 1 l . In this language of quantum channels, the depolarizing channel E DP with error parameter p acting on any one-qubit quantum state ρ ∈ C 2 × 2 as It was observed that when the temperature of a physical system is zero or very low, the error A † 1 is actually negligible [22]. For simplicity, we further ignore the phase damping error (which is characterized by the Pauli operator Z ). Then the error model is fully characterized by A$_{0}$ and A$_{1}$ . In this work, we will focus on this quantum channel with only amplitude damping noise, i.e. the AD channel E $_{AD}$, with only two Kraus operators given by Eq. (4). The AD channel is the simplest nonunital channel whose Kraus operators cannot be described by Pauli operations. The AD channel is a quantum analogue of the classical Z -channel which transmits 0 faithfully, but maps so the Kraus operators for the depolarizing channel are the Pauli matrices together with identity. However, if further information about an error process is available, more efficient codes can be designed. Indeed in many physical systems, the types of noise are likely to be unbalanced between amplitude ( X -type) errors and phase ( Z - type) errors. Recently a lot of attention has been put into arXiv:1001.2356v1 [quant-ph] 14 Jan 2010 I. INTRODUCTION E $_{AS}$( ρ ) = (1 − (2 p$_{x}$ + p$_{z}$ )) ρ + p$_{x}$ ( XρX + Y ρY ) + p$_{z}$ZρZ. (3) A$_{0}$ = ( 1 0 0 √$_{1}$ − γ ) and A$_{1}$ = ( 0 √ γ 0 0 ) . (4) A$_{1}$ = ( 0 √ γ 0 0 ) = √ γ 2 ( X + iY ) A † 1 = √ γ 2 ( X − iY ) . Note that and N ( ρ ) = ∑ k A$_{k}$ρA † $_{k}$, (1) E $_{DP}$( ρ ) = (1 − p ) ρ + p 3 ( XρX + Y ρY + ZρZ ) , (2)