Encoding one logical qubit into six physical qubits

Abstract

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum errorcorrecting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a CalderbankShor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.

Dualities and identities for entanglement-assisted quantum codes

“…The case where noise occurs on the ebits was considered in Refs. [31,40,21]. The simplified stabilizer subgroup S ofḠ n is S = g k+1 , · · · , g k+c , h k+1 , · · · , h k+c , g k+c+1 , · · · , g n .…”

Quantum convolutional coding with shared entanglement: general structure

“…Detailed algorithms exist to find an encoding circuit-please see Refs. [24,23,10,28]. These algorithms can also handle a set of generators that have more complicated commutation relations.…”

“…A general set of generators for a quantum block code can have complicated commutation relations. There exists a symplectic Gram-Schmidt orthogonalization procedure that simplifies the commutation relations [24,23,10,28]. Specifically, the algorithm performs row operations that do not affect the code's error-correcting properties and thus gives a set of generators that form an equivalent code.…”

“…The focus of the next two sections is to elucidate techniques for reducing a set of quantum convolutional generators to have the standard commutation relations given above. The difference between the current techniques and the former techniques [24,23,10,28] is that the current techniques operate on generators that have a convolutional form rather than a block form. The next section shows how to expand a set of quantum convolutional generators to simplify their commutation relations.…”

Gauge subsystems, separability and robustness in autonomous quantum memories

“…In figure 2 we display the average fidelity decay F(t) ≡ 0 |ρ t | 0 over 10 4 simulated quantum trajectories (each) for lossy quantum memories implementing the five-, six-, sevenand nine-qubit codes [20,22,23], which may be compared directly with analogous results from our prior work on other codes [6][7][8]. In figure 2 the nine-qubit code results are shown for the loss-tolerant circuit layout; analogous simulations for the non-tolerant layout of the nine-qubit code are presented in figure 7 of [8] and are very similar to the results shown for the fivequbit code here.…”