Suggested reading
For the basis of the tableaux methods first read (Gottesman, 1998) followed by the more efficient approach described in (Aaronson and Gottesman, 2004).
The tableaux can be canonicalized (i.e. Gaussian elimination can be performed on them) in a number of different ways, and considering the different approaches provides useful insight. The following methods are implemented in this library:
- The default one: (Garcia et al., 2012)
- Useful when in need of tracing out a set of qubits: (Audenaert and Plenio, 2005)
- Useful when defining logical operators of codes: (Gottesman, 1997)
For the use of these methods in error correction and the subtle overlap between the two fields consider these resources. They are also useful in defining some of the specific constraints in commutation between rows in the tableaux:
These publications describe the uniform sampling of random stabilizer states:
For circuit construction routines (for stabilizer measurements for a given code):
- (Cleve and Gottesman, 1997)
- (Gottesman, 1997) (and its erratum)
- (Grassl, 2002)
- (Grassl, 2011)
For quantum code construction routines:
- (Cleve and Gottesman, 1997)
- (Gottesman, 1996)
- (Gottesman, 1997)
- (Yu et al., 2013)
- (Chao and Reichardt, 2017)
- (Kitaev, 2003)
- (Fowler et al., 2012)
- (Knill and Laflamme, 1996)
- (Steane, 1999)
- (Campbell et al., 2012)
- (Anderson et al., 2014)
For classical code construction routines:
- (Muller, 1954)
- (Reed, 1954)
- (Raaphorst, 2003)
- (Abbe et al., 2020)
- (Djordjevic, 2021)
- (Hocquenghem, 1959)
- (Bose and Ray-Chaudhuri, 1960)
- (Bose and Ray-Chaudhuri, 1960)
- (Lin and Costello, 2024)
References
- Aaronson, S. and Gottesman, D. (2004). Improved simulation of stabilizer circuits. Physical Review A 70, 052328.
- Abbe, E.; Shpilka, A. and Ye, M. (2020). Reed–Muller codes: Theory and algorithms. IEEE Transactions on Information Theory 67, 3251–3277.
- Anderson, J. T.; Duclos-Cianci, G. and Poulin, D. (2014). Fault-tolerant conversion between the steane and reed-muller quantum codes. Physical review letters 113, 080501.
- Audenaert, K. M. and Plenio, M. B. (2005). Entanglement on mixed stabilizer states: normal forms and reduction procedures. New Journal of Physics 7, 170.
- Bose, R. C. and Ray-Chaudhuri, D. K. (1960). Further results on error correcting binary group codes. Information and Control 3, 279–290.
- Bose, R. C. and Ray-Chaudhuri, D. K. (1960). On a class of error correcting binary group codes. Information and control 3, 68–79.
- Bravyi, S. and Maslov, D. (2021). Hadamard-free circuits expose the structure of the Clifford group. IEEE Transactions on Information Theory 67, 4546–4563.
- Brown, W. and Fawzi, O. (Jul 2013). Short Random Circuits Define Good Quantum Error Correcting Codes. In: 2013 IEEE International Symposium on Information Theory; pp. 346–350.
- Calderbank, A. R.; Rains, E. M.; Shor, P. and Sloane, N. J. (1998). Quantum error correction via codes over GF (4). IEEE Transactions on Information Theory 44, 1369–1387.
- Campbell, E. T.; Anwar, H. and Browne, D. E. (2012). Magic-state distillation in all prime dimensions using quantum reed-muller codes. Physical Review X 2, 041021.
- Chao, R. and Reichardt, B. W. (2017). Quantum Error Correction with Only Two Extra Qubits. Physical review letters 121 5, 050502.
- Cleve, R. and Gottesman, D. (1997). Efficient computations of encodings for quantum error correction. Physical Review A 56, 76.
- Djordjevic, I. B. (2021). Quantum information processing, quantum computing, and quantum error correction: an engineering approach (Academic Press).
- Fowler, A. G.; Mariantoni, M.; Martinis, J. M. and Cleland, A. N. (2012). Surface codes: Towards practical large-scale quantum computation. Physical Review A 86, 032324.
- Garcia, H. J.; Markov, I. L. and Cross, A. W. (2012). Efficient inner-product algorithm for stabilizer states, arXiv preprint arXiv:1210.6646.
- Gottesman, D. (1996). Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A 54, 1862.
- Gottesman, D. (1997). Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology.
- Gottesman, D. (1998). The Heisenberg representation of quantum computers. In: International Conference on Group Theoretic Methods in Physics (Citeseer).
- Grassl, M. (2002). Algorithmic aspects of quantum error-correcting codes. Mathematics of Quantum Computation, 223–252.
- Grassl, M. (2011). Variations on encoding circuits for stabilizer quantum codes. In: International Conference on Coding and Cryptology (Springer); pp. 142–158.
- Gullans, M. J.; Krastanov, S.; Huse, D. A.; Jiang, L. and Flammia, S. T. (2021). Quantum Coding with Low-Depth Random Circuits. Physical Review X 11, 031066.
- Hocquenghem, A. (1959). Codes correcteurs d'erreurs. Chiffers 2, 147–156.
- Kitaev, A. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics 303, 2–30.
- Knill, E. and Laflamme, R. (1996). Concatenated quantum codes, arXiv preprint quant-ph/9608012.
- Koenig, R. and Smolin, J. A. (2014). How to efficiently select an arbitrary Clifford group element. Journal of Mathematical Physics 55, 122202.
- Krastanov, S.; de la Cerda, A. S. and Narang, P. (2020). Heterogeneous Multipartite Entanglement Purification for Size-Constrained Quantum Devices, arXiv preprint arXiv:2011.11640.
- Li, Y.; Chen, X. and Fisher, M. P. (2019). Measurement-driven entanglement transition in hybrid quantum circuits. Physical Review B 100, 134306.
- Lin, S. and Costello, D. (2024). Error Control Coding (Pearson).
- MacKay, D. J.; Mitchison, G. and McFadden, P. L. (2004). Sparse-graph codes for quantum error correction. IEEE Transactions on Information Theory 50, 2315–2330.
- Muller, D. E. (1954). Application of Boolean algebra to switching circuit design and to error detection. Transactions of the IRE professional group on electronic computers, 6–12.
- Nahum, A.; Ruhman, J.; Vijay, S. and Haah, J. (2017). Quantum Entanglement Growth under Random Unitary Dynamics. Physical Review X 7, 031016.
- Panteleev, P. and Kalachev, G. (2021). Degenerate Quantum LDPC Codes With Good Finite Length Performance. Quantum 5, 585, arXiv:1904.02703 [quant-ph].
- Panteleev, P. and Kalachev, G. (Jun 2022). Asymptotically Good Quantum and Locally Testable Classical LDPC Codes. In: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (ACM, Rome Italy); pp. 375–388.
- Raaphorst, S. (2003). Reed-muller codes. Carleton University, May 9.
- Raveendran, N.; Rengaswamy, N.; Rozpędek, F.; Raina, A.; Jiang, L. and Vasić, B. (2022). Finite Rate QLDPC-GKP Coding Scheme That Surpasses the CSS Hamming Bound. Quantum 6, 767.
- Reed, I. S. (1954). A class of multiple-error-correcting codes and the decoding scheme. IEEE Transactions on Information Theory 4, 38–49.
- Roffe, J.; Cohen, L. Z.; Quintavalle, A. O.; Chandra, D. and Campbell, E. T. (2023). Bias-Tailored Quantum LDPC Codes. Quantum 7, 1005.
- Steane, A. M. (1999). Quantum reed-muller codes. IEEE Transactions on Information Theory 45, 1701–1703.
- Steane, A. M. (2007). A tutorial on quantum error correction. In: PROCEEDINGS-INTERNATIONAL SCHOOL OF PHYSICS ENRICO FERMI, Vol. 162 (IOS Press; Ohmsha; 1999); p. 1.
- Van Den Berg, E. (2021). A simple method for sampling random Clifford operators. In: 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE); pp. 54–59.
- Wilde, M. M. (2009). Logical operators of quantum codes. Physical Review A 79, 062322.
- Yu, S.; Bierbrauer, J.; Dong, Y.; Chen, Q. and Oh, C. H. (2013). All the Stabilizer Codes of Distance 3. IEEE Transactions on Information Theory 59, 5179–5185.