This paper introduces a novel abstraction for programming quantum operations, specifically projective Cliffords , as functions over the qu d it Pauli group. Generalizing the idea behind Pauli tableaux, we introduce a type system and lambda calculus for projective Cliffords called λ P c that captures well-formed Clifford operations via a Curry-Howard correspondence with a particular encoding of the Clifford and Pauli groups. In λ P c , users write functions that encode projective Cliffords P ↦ UPU † , and such functions are compiled to circuits executable on modern quantum computers that transform quantum states | φ ⟩ into U | φ ⟩, up to a global phase. Importantly, the language captures not just qubit operations, but qu d it operations for any dimension d . Throughout the paper we explore what it means to program with projective Cliffords through a number of examples and a case study focusing on stabilizer error correcting codes.
In this work, we compute the number of [[n,k]]d stabilizer codes made up of d -dimensional qudits, for arbitrary positive integers d . In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm , but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \cite{Gross2006} when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.
Different quantum error correction schemes trade off overhead, error suppression, and hardware connectivity. Code concatenation can relax these tradeoffs by using an outer code whose non-local connectivity is supplied by logical operations of an inner code rather than directly by hardware. Prior works showed that this can reduce memory overhead for local low-rate inner codes such as the surface code. Here, we study concatenation over non-local, high-rate inner codes. Such inner codes experience correlated errors among the many logical qubits in a single codeblock. We handle this by treating each block as a single logical Galois qudit, enabling concatenation with algebraic outer codes with excellent parameters and, crucially, list decoders. In particular, we consider a memory system formed by concatenating quantum Reed-Solomon outer codes over the gross code. For fault-tolerant syndrome extraction, we develop a Galois qudit Shor scheme using "time-like" Reed-Solomon protection against measurement errors. Interestingly, a lightweight fault tolerance scheme, that would fail for qubits, works well for large-alphabet qudits, suggesting a very different theory of fault tolerance for such qudits. The whole protocol is optimised via improved bicycle instruction logical error rates, novel compilation strategies, and recent decoder post-selection rules. At uniform
10−3 physical noise, the concatenated gross code reaches the teraquop regime, which it previously could not access, with a lower space overhead than the
288-qubit two-gross code, while offering several advantages from the engineering standpoint. Beyond our main case study, we believe the core ideas of Galois qudits, quantum Reed-Solomon outer codes, and list decoding, will prove generically powerful and highly transferable ideas across high-rate quantum architectures.
Qudits with local dimension d>2 can have unique structure and uses that qubits ( d=2 ) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d , we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.
The qubit has become so dominant in quantum computing that alternatives are rarely discussed, yet it is a historical accident: two-level systems were simply the easiest to isolate experimentally. This paper argues that qudits —
d-dimensional quantum systems with
d>2 — offer strictly more across every metric: error-correction headroom, richer topological structure, sharper computational phase transitions, and a natural home for the exceptional algebras that govern the deepest structures in mathematics and physics. Two values of
d are mathematically privileged:
d=4 (spacetime spinors, two-qubit entanglement, quaternionic structure) and
d=8 (octonions,
G2, the Fano plane). These are consequences of Hurwitz's theorem (1898), which identifies
d∈1,2,4,8 as the only dimensions supporting normed division algebras. The paper covers: the
SU(4) containment that makes two-qubit logic native to a single ququart; Dirac spinors as the natural
d=4 Hilbert space; the
d=8 octonion layer and its
G2 symmetry; topological advantages of higher
d including Hopfion braiding and non-Abelian statistics; and the Hopf fibration tower
S1→S3→S2→⋯→S7→S15→S8 as a construction manual for building qudits from qubits. v2.0 adds five new developments: The padit tower. The
p-adic qudit (padit) in
L2(Z/pnZ) extends Hurwitz to all prime powers
d=pn. Qubits are the
p=2,
n=1 special case. The Number-Theoretic Transform is the exact
p-adic QFT (no floating-point error). The Fano plane is the
p=2,
n=3 padit — the natural home of the 3-qubit Origami ISA and the Steane
[![7,1,3]!] code. Bredon cohomology cascade. Equivariant cohomology of the Weyl chamber proves
χBredon=2 (2-qubit, one topological class distinguishing CNOT from iSWAP)
→0 (3-qubit, equivariantly contractible)
→ non-trivial (7-qubit Fano). The 3-qubit native gate architecture (TriQ) adds no topological value; the correct jump is directly from 2-qubit to Steane-7. FeMoco as a room-temperature
d=8 computer. The iron-molybdenum cofactor of nitrogenase contains 7 Fe atoms corresponding to the 7 points of
PG(2,2). The
N2 fixation reaction is a 14-opcode Fano programme running at 300 K, decoherence-free in the
G2-symmetric Clifford sector. Nature builds
d=8 qudit computers; we merely need to recognise them. Shor's algorithm has zero magic. The magic resource theory shows that Shor's algorithm achieves exponential speedup with mana
=0: all gates lie in the Clifford sector. Qudits have more Clifford structure, not more overhead. The standard objection — that higher
d requires more magic-state distillation — is precisely backwards. The architecture ladder. The Bredon cascade, the Hopf fibration tower, and the padit tower converge on the same four-level stack: native 2-qubit gates (
d=4,
χBredon=2)
→ Steane
[![7,1,3]!] logical qudits (
d=8, Fano)
→ compiled Toffoli and above. The open problem of v1 (design a
G2-symmetric Fano QEC code) is resolved: the Steane code is that code. Third, molecular evidence: the FeMo-cofactor [Mo–7Fe–9S–C–homocitrate] of nitrogenase is a 7-qubit Fano computer running at 300 K, room temperature, decoherence-free in the
G2-symmetric Clifford sector. The 7 Fe atoms occupy the 7 points of
PG(2,2); the molybdenum apex and the interstitial carbide carbon (Spatzal et al. 2011, Science) provide structural scaffolding but do not carry Fano-register qubits. Nature builds
d=8 qudit computers; we merely need to recognise them.
This is a note from a series of lectures at Encuentro Colombiano de Computacion Cuantica, Universidad de los Andes, Bogota, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Clifford quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Clifford circuits.