We provide a geometric classification of single-qubit projective measurements on one-dimensional linear cluster states within a topological framework. Establishing an explicit correspondence between local measurements and surgery operations on an associated link model, we represent the cluster state as a linear Hopf chain. Computational-basis (Z) measurements act as topological severance for bulk qubits and boundary pruning for end qubits. Transverse-basis (X) measurements remove the measured qubit and stratify the remaining state into a superposition of two disjoint but classically correlated segments. In contrast, lateral-basis (Y) measurements preserve a single continuous spliced chain while generating complex phase factors absent from unframed link descriptions. Although the unframed linking structure already distinguishes X- and Y-basis measurement outcomes geometrically, it cannot differentiate the two possible outcomes within a fixed basis. To resolve this ambiguity, we introduce a framed ribbon representation in which quantum phases are encoded as geometric twists, with chiral plus/minus 90 deg twists representing the phases plus/minus i. The resulting framework provides a phase-sensitive, outcome-resolved geometric description of single-shot Pauli measurements on linear cluster states. The twist angles are geometric labels determined by measurement outcomes and associated by-product operators rather than topological invariants, whereas the underlying linking pattern remains a genuine topological datum. The analysis is restricted to single-shot single-qubit Pauli measurements on one-dimensional cluster states; sequential measurements and classical feedforward are left as open problems.
We provide a geometric classification of single-qubit projective measurements on one-dimensional linear cluster states within a topological framework. Establishing an explicit correspondence between local measurements and surgery operations on an associated link model, we represent the cluster state as a linear Hopf chain. Computational-basis (Z) measurements act as topological severance for bulk qubits and boundary pruning for end qubits. Transverse-basis (X) measurements remove the measured qubit and stratify the remaining state into a superposition of two disjoint but classically correlated segments. In contrast, lateral-basis (Y) measurements preserve a single continuous spliced chain while generating complex phase factors absent from unframed link descriptions. Although the unframed linking structure already distinguishes X- and Y-basis measurement outcomes geometrically, it cannot differentiate the two possible outcomes within a fixed basis. To resolve this ambiguity, we introduce a framed ribbon representation in which quantum phases are encoded as geometric twists, with chiral plus/minus 90 deg twists representing the phases plus/minus i. The resulting framework provides a phase-sensitive, outcome-resolved geometric description of single-shot Pauli measurements on linear cluster states. The twist angles are geometric labels determined by measurement outcomes and associated by-product operators rather than topological invariants, whereas the underlying linking pattern remains a genuine topological datum. The analysis is restricted to single-shot single-qubit Pauli measurements on one-dimensional cluster states; sequential measurements and classical feedforward are left as open problems.
We consider a restricted two-qubit model in which an unknown local unitary acts on one subsystem of a maximally entangled qubit pair and only local Pauli-product correlations are measured. We show that the three same-axis correlators XX, YY, and ZZ determine the magnitudes of the four components of the associated unit quaternion but, at generic interior points, leave an eightfold ambiguity in the underlying local rotation. We express this ambiguity as a sign fiber isomorphic to F2^3. The six cross-axis Pauli correlators separate into three rank-two sign-character subspaces. This structure yields a complete classification of the fifteen possible pairs of cross-axis correlators: the three reciprocal pairs XY/YX, XZ/ZX, and YZ/ZY remain rank deficient and generically leave a twofold ambiguity, whereas the remaining twelve pairs generically resolve the eightfold alias under explicit amplitude-nondegeneracy conditions. We then formulate a finite-shot adaptive reconstruction procedure that selects well-separated correlator groups and abstains near degeneracy, and we distinguish conditional reconstruction from held-out model auditing. The results concern identifiability within a restricted local-unitary model and do not constitute a new general tomography bound or evidence for additional physical states.
The quantum measurement problem asks how the smooth, unitary evolution of a superposition gives way to a single, definite outcome when a macroscopic detector interacts with a quantum system. This paper presents a geometric toy model of this interaction, with a deterministic factorization stage and a probabilistic selection step. The entangled state of a quantum system and a detector is embedded into semi-structured complex (ℍ) state space, where newly introduced projection operators, called trigger transformations, factorize the entangled state into a tensor product of two identical subsystems: one describing the quantum system and one describing what the detector registers. This mirrored factorization is shown to reduce exactly to the original entangled state on projection back to standard Hilbert space. Branch-indexed selector operators R, also introduced here, then return the state to Hilbert space and retain a single branch (or measurement outcome), with the standard quantum probabilities acting as structural weights of the branches. The model thereby confines the non-unitary step of measurement to a single, explicitly defined selection operation, offering an alternative to an ad-hoc wave function collapse. In its present form, the outcome rule and the Born weights are stated as an explicit postulate rather than derived, and a possible coupling to background vacuum fluctuations is identified as a route to future experimental tests.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.
In multi-qubit quantum control, target unitary operations are commonly specified through full-unitary target descriptions and assessed through global comparison measures. In this work, we introduce an interaction-resolved decomposition of n-qubit unitaries that provides explicit access to their many-body interaction structure through computational-basis phases collected in a diagonalizing frame. Such a frame is conveniently given by local rotations for many operationally relevant operations, including gates generated by single Pauli strings or commuting sets of Pauli strings, such as stabilizer operations, controlled-phase gates, Toffoli-type operations, and Ising interactions. We derive parity-weighted sums of these computational-basis phases that exactly and uniquely resolve k-body interaction terms supported on arbitrary qubit subsets, which we term support-selective phase invariants. These invariants provide an interaction-resolved coordinate system that organizes unitary operations according to their multipartite interaction structure, giving direct access to local, pairwise, tripartite, and general k-partite interaction content underlying entangling operations. This enables the formulation of selective quantum optimal control targets for synthesizing desired combinations of many-body interactions. We supplement this with numerical demonstrations for a representative hardware model, a realistic nitrogen-vacancy spin register, where we synthesized isolated tripartite interactions up to local equivalence within a single control pulse, guided by these invariants, for both diagonal (ZZZ) and non-diagonal (XZZ) terms.
Abstract Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X , Y , and Z operators. This novel computational feature is related to the ‘genuine’ 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity.
Standard quantum information is formulated over complex Hilbert spaces, where the pure state space of a single qubit is geometrically encoded by the first Hopf fibration S3→S2. Beyond this familiar setting, the normed division algebras C, H and O provide a hierarchy of increasingly rich algebraic and topological structures that has motivated several extensions of quantum theory and quantum computation. This review synthesizes the literature connecting quaternionic and octonionic frameworks, Clifford algebras, spinors, projective spaces and Hopf fibrations. We emphasize a central conceptual point that is often blurred in the literature: the second and third Hopf fibrations play two distinct roles, namely, (i) as kinematical descriptions of hypercomplex single-particle state spaces such as the quaternionic projective line HP1≃S4, and (ii) as entanglement-sensitive descriptions of multi-qubit complex systems, especially two- and three-qubit Hilbert spaces. On the algebraic side, Clifford and geometric algebras provide a natural language for rotations, spinors, and gate synthesis, while quaternionic Hilbert modules furnish a mathematically consistent extension of standard qubit kinematics and dynamics. By contrast, octonionic models face major obstructions due to non-associativity, which affects inner products, tensor products, spectral theory and circuit composition. We therefore distinguish carefully between robust results, partial constructions and speculative directions. The outcome is a unified geometric review of hypercomplex quantum information, together with a map of open problems at the interface of topology, noncommutative algebra, and quantum computation.