Memory system using a quantum convolutional code
A memory system comprising a qubit array configured to store therein one or more entangled qubit states encoded using a quantum stabilizer code. The memory system further comprises a quantum-state-refresh module configured to refresh an entangled qubit state in the qubit array when a degradation error is detected therein. The quantum-state-refresh module is further configured to detect the degradation error by performing a redundant measurement of a set of syndrome values corresponding to the quantum stabilizer code. The redundant measurement is based on an error-correction code defined using the generator matrix of the quantum stabilizer code and a corresponding supplemental parity-check matrix. In an example embodiment, each of the generator and supplemental parity-check matrices has a respective inclined-stripe form.
1. An apparatus comprising:
a register configured to store a coded entangled qubit state encoded using a quantum stabilizer code;
a measurement circuit configured to perform a redundant measurement of a set of syndrome values corresponding to the coded entangled qubit state to generate a measurement vector having the set of syndrome vlaues and a set of parity values, the redundant measurement being performed based on a convolutinal error-correction code defined using an ordered set of generator vectors of the quantum stabilizer code and a plurality of linearly independent combinations of the generator vectors; and
an electronic decoder configured to use the convolutional error-correction code to correct a measurement error in the set of syndrome values in the measurement vector on which the redundant measurement has been performed; and
wherein:
the ordered set of generator vectors are defined by coefficients in a generator matrix having a first inclined-stripe form; and
the plurality of linearly independent combinations of the generator vectors are defined by coefficients in a supplemented parity-check matrix having a second inclinded-stripe form.
2. The apparatus of claim 1 , wherein the second inclined-stripe form is a transposed form with respect to the first inclined-stripe form.
3. The apparatus of claim 1 . wherein each of the coefficients in the supplemental parity-check matrix has one of two binary values.
4. The apparatus of claim 1 , wherein the second inclined-stripe form is constructed by incrementally shifting down a continuous subset of topmost matrix elements of a leftmost column of the supplemental parity-check matrix.
5. The apparatus of claim 1 , wherein the generator matrix is a matrix over a Galois field.
6. The apparatus of claim 1 , wherein the generator matrix is a matrix over a Galois field of size 4.
7. The apparatus of claim 1 , wherein the first inclined-stripe form is constructed by incrementally shifting right a continuous sub-block of leftmost matrix elements of a topmost row block of the generator matrix.
8. The apparatus of claim 1 , wherein the electronic decoder is configured to correct the measurement error based on trellis-type decoding.
9. The apparatus of claim 1 , further comprising a state-recovery circuit configured to correct a degradation error in the coded entangled qubit state based on the set of syndrome values received from the electronic decoder.
10. The apparatus of claim 9 , wherein the state-recovery circuit is further configured to:
determine, based on the set of syndrome values received from the electronic decoder, an error operator corresponding to a most-probable error in the coded entangled qubit state; and
subject the coded entangled qubit state to a stimulus determined based on the error operator.
11. The apparatus of claim 9 , further comprising an array for storing a plurality of coded entangled qubit states, wherein the apparatus is configured to:
transfer a selected coded entangled qubit state from the array to the register; and
transfer the selected coded entangled qubit state from the register to the array after the degradation error is corrected.
12. The apparatus of claim 1 , wherein the measurement circuit comprises:
a set of first measurement modules coupled to the register, each of the first measurement modules being configured to measure a respective syndrome value of the coded entangled qubit state based on a respective one of the generator vectors;
a set of second measurement modules coupled to the register, each of the second measurement modules being configured to measure a respective parity value corresponding to the syndrome of the coded entangled qubit state based on a respective one of the linearly independent combinations of the generator vectors.
13. The apparatus of claim 1 , wherein a measurement module in the measurement circuit comprises:
a plurality of quantum gates connected to the register and configured to process a reference multi-qubit state by coupling a subset of qubits of the coded entangled qubit state to said reference multi-qubit state; and
a logic circuit configured to estimate a syndrome value or a parity value of the coded entangled qubit state from measurements on the processed reference multi-qubit state performed using the quantum gates.