Quantum codes in the Lee metric

Real quantum hardware has far more than two levels per particle, from the tower of states of a superconducting circuit to the many internal states of an atom or ion, yet most quantum codes use only two of them. Codes on q-level qudits could ease the challenge of scaling up. However, existing qudit codes are usually judged by how many qudits an error touches, as if all possible errors on a qudit were equally likely.

That is not realistic. On a large spin, such as a nuclear spin, the dominant errors are small shifts of the state by one or two steps. These are the discrete cousins of the small displacements that GKP codes for oscillators are built to correct. The Lee metric, long used in classical coding theory, measures an error by the total size of its shifts. That makes it the natural yardstick for this kind of noise.

In a collaboration between Andrew Lucas’s group at the University of Colorado Boulder and Victor Albert’s group at QuICS, we developed stabilizer and CSS codes over ℤ_q for any q. We measured them in two Lee metrics: one treats the X and Z shifts of an error separately, and the other adds them. Even for qubits this changes things, because the joint metric counts a Y error twice, which is the right accounting when bit flips and phase flips come from independent processes. It allows a 9-qubit code that corrects any two X or Z errors or any single Y error, whereas correcting any two errors in the usual Hamming metric takes at least 11 qubits.

We built families of such codes, including lifted Hamming and Golay codes obtained from classical self-dual codes with the construction of our earlier work on transversal gates. We also studied fault tolerance in this setting, including how Clifford gates spread the size of errors. A quantum version of the Gray map turns codes on four-level qudits into qubit codes, some with two-fold transversal gates beyond the qubit Clifford group.

Two results set the limits. For CSS Lee-LDPC codes on n qudits, the Lee distance can grow at most linearly in n, however large q is, so the big internal space of each qudit does not by itself make for a high-distance code. On the other hand, certain classical “helical repetition codes” on large qudits keep their memory for exponentially long times at fixed temperature, even in one dimension, once q grows with the system size. Their hypergraph products give local two-dimensional quantum codes that inherit this self-correction against Z errors; whether they also self-correct against X errors is open.