Store a quantum state reliably for a macroscopic

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Store a quantum state reliably for a macroscopic time in a presence of hardware imperfections and thermal noise without active error correction.

Towards topological self-correcting memories Most promising ideas:

Qubits live at sites of a 2 D or 3 D lattice. O(1) qubits per site. Hamiltonian = sum of local commuting Pauli stabilizers

Example: 3 D Cubic Code [Haah 2011 ]

Stabilizer code Hamiltonians with TQO: previous work

Must be local, trace preserving, completely positive Each qubit is acted on by O(1) Lindblad operators.

Heat bath Memory system Lindblad operator transfers energy the system to the bath (quantum jump). The spectral density from obeys detailed balance:

A list of all measured eigenvalues is called a syndrome. Error correction algorithm The net action of the decoder: is the projector onto the subspace with syndrome s

Defect = spatial location of a violated stabilizer, Defect diagrams will be used to represent syndromes. Example: 2 D surface code: 1 2 4 3 X-error Z-error decoder’s task is to annihilate the defects in a way which is most likely to return the memory to its original state.

1 2 3 4 5

1 2

The decoder stops whenever all defects have been annihilated, or when the unit of length reached the lattice size. The correcting operator is chosen as the product of all recorded annihilation operators. Failure 1: decoder has reached the maximum unit of length, but some defects are left. Failure 2: all defects have been annihilated but the correcting operator does not return the system to the original state. RG decoder can be implemented in time poly(L)

Derive an upper bound on the worst-case storage error:

The theorem only provides a lower bound on the memory time. Is this bound tight ? Monte-Carlo simulation We observed the exponential decay: Numerical estimate the memory time:

Numerical test of the scaling

Main theorem: sketch of the proof

Some terminology An error path implementing a Pauli operator P is a finite sequence of single-qubit Pauli errors whose combined action coincides with P. Energy cost = maximum number of defects along the path. Energy barrier of a Pauli operator P is the smallest integer m such that P can be implemented by an error path with energy cost m

Basic intuition behind self-correction: The thermal noise is likely to generate only errors with a small energy barrier. Decoder must be able to correct them. Errors with high energy barrier can potentially confuse the decoder. However, such errors are not likely to appear.

Suppose we choose and Then the entropy factor can be neglected: In order to have a non-trivial bound, we need at least logarithmic energy barrier for all uncorrectable errors:

More terminology [Haah 2011] A logical string segment is a Pauli operator whose action on the vacuum creates two well-separated clusters of defects. vacuum The smallest cubic boxes enclosing the two clusters of defects are called anchors

More terminology A logical string segment is trivial iff its action on the vacuum can be reproduced by operators localized near the anchors: vacuum

There exist a constant α such that any logical string segment with aspect ratio > α is trivial. Aspect ratio = Distance between the anchors Size of the acnhors 3 D Cubic Code obeys the no-strings rule with α=15 [Haah 2011] No 2 D stabilizer code obeys the no-strings rule [S. B. , Terhal 09]

S S’

S S’

e e sparse e 0 1 2 3 4

Renormalization group method

No-strings rule can be used to `localize’ level-p errors by multiplying them by stabilizers. Localized level-p errors connecting syndromes S and S’ act on r(p)-neighborhood of S and S’.

At least one syndrome must be dense at all levels. Such syndrome must contain at least log(L) defects.

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