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CRC Press eBooks

Quantum Circuits

Frank Gaitan

Published Feb 7, 2008• 0 citations
Publisher

Abstract

In this final chapter we present a number of bounds on the minimum dis­ tance of a quantum error correcting code (QECC). Sections 7.1-7.3 derive the quantum Hamming, Gilbert-Varshamov, and Singleton bounds, respectively. The corresponding classical bounds were derived in Section 1.2.5. Section 7.4 introduces quantum generalizations of the classical MacWilliams identities, and of weight and shadow enumerators [1,2]. With these tools in hand, the problem of finding an optimal upper bound on the code distance is reduced to solving a problem in linear programming [3-5]. Finally, we close in Section 7.5 by showing how QECCs can be connected to the process of entanglement pu­ rification [6,7]. This final section allows our study of QECCs to segue into the developing field of quantum information theory [8,9], and provides a good place to bring the discussion in this book to a close.

DOI
10.1201/b15868-11

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In this final chapter we present a number of bounds on the minimum dis­ tance of a quantum error correcting code (QECC). Sections 7.1-7.3 derive the quantum Hamming, Gilbert-Varshamov, and Singleton bounds, respectively. The corresponding classical bounds were derived in Section 1.2.5. Section 7.4 introduces quantum generalizations of the classical MacWilliams identities, and of weight and shadow enumerators [1,2]. With these tools in hand, the problem of finding an optimal upper bound on the code distance is reduced to solving a problem in linear programming [3-5]. Finally, we close in Section 7.5 by showing how QECCs can be connected to the process of entanglement pu­ rification [6,7]. This final section allows our study of QECCs to segue into the developing field of quantum information theory [8,9], and provides a good place to bring the discussion in this book to a close.
Abstract—The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from ex-changing the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglement-assisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel. Index Terms—quantum dual code, entanglement-assisted quantum error correction, MacWilliams identity, linear programming bound, entanglement-assisted repetition codes, entanglement-assisted accumulator codes I.
We saw in Chapter 1 how the 3-fold repetition code would drastically reduce the probability of messages being wrongly decoded, and discussed the price which this entailed. Then in Chapter 2 we saw how the Hamming code solved the same problem of guaranteeing to correct every instance of a single error per codeword at much less cost. The way in which it achieved its impressive performance was by ensuring that amongst the 16 7-bit codewords, no pair of them was separated by a Hamming distance less than 3. We also developed general results, Theorems 2.1 and 2.2, to connect the minimum distance of the code with its error detecting and correcting potential.
Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.
As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.