Can an encoded quantum state be controlled continuously without repeated measurements to prevent leakage? This study gives a constructive answer for a four-spin-2 AKLT plaquette, where a three-dimensional singlet code sits inside a 625-dimensional physical Hilbert space. The result combines an exact restriction with a coherent control scheme: strictly leakage-free two-site Hamiltonians cannot provide universal control of the full encoded qutrit, yet two explicitly constructed interactions enable universal single-qubit control without intermediate measurements or postselection within a two-dimensional subspace. The classification begins with the complete physical leakage map for all six spin pairs and exchange polynomials through degree four. Exact rational calculations establish rank L = 6 and dim ker L = 18 for the 255 × 24 matrix. Despite this large family of leakage-cancelling combinations, every accessible logical Hamiltonian obeys [A, R] = 0, where R is a fixed geometric involution. Its two-dimensional and one-dimensional eigenspaces therefore remain separate, obstructing universal SU(3) control. The restriction extends to arbitrary anisotropic two-site Hamiltonians that preserve the code. Within the rotationally invariant pair-polynomial model, interactions confined to the four plaquette edges act only as logical scalars; noncommuting protected controls require at least three pair supports and polynomial degree three. The constructive solution uses quadratic and cubic exchange Hamiltonians acting simultaneously on three pairs, including a diagonal link. Their coefficients cancel physical leakage exactly, so QH₂P = QH₃P = 0 throughout the full system, with P projecting onto the code and Q onto its complement. Their logical generators satisfy Lie(iA, iB) ≅ u(2) ⊕ u(1), allowing arbitrary coherent qubit gates while preserving the code at every intermediate time. The manuscript supplies exact certificates and verification procedures, together with operator-norm and calibration-error bounds that make the engineering requirements explicit. The practical opportunity is to remove intermediate code-check measurements from this control scheme; demonstrating a hardware advantage requires calibrated simultaneous interactions, including the diagonal channel, and measured gate fidelity and latency.
Can an encoded quantum state be controlled continuously without repeated measurements to prevent leakage? This study gives a constructive answer for a four-spin-2 AKLT plaquette, where a three-dimensional singlet code sits inside a 625-dimensional physical Hilbert space. The result combines an exact restriction with a coherent control scheme: strictly leakage-free two-site Hamiltonians cannot provide universal control of the full encoded qutrit, yet two explicitly constructed interactions enable universal single-qubit control without intermediate measurements or postselection within a two-dimensional subspace. The classification begins with the complete physical leakage map for all six spin pairs and exchange polynomials through degree four. Exact rational calculations establish rank L = 6 and dim ker L = 18 for the 255 × 24 matrix. Despite this large family of leakage-cancelling combinations, every accessible logical Hamiltonian obeys [A, R] = 0, where R is a fixed geometric involution. Its two-dimensional and one-dimensional eigenspaces therefore remain separate, obstructing universal SU(3) control. The restriction extends to arbitrary anisotropic two-site Hamiltonians that preserve the code. Within the rotationally invariant pair-polynomial model, interactions confined to the four plaquette edges act only as logical scalars; noncommuting protected controls require at least three pair supports and polynomial degree three. The constructive solution uses quadratic and cubic exchange Hamiltonians acting simultaneously on three pairs, including a diagonal link. Their coefficients cancel physical leakage exactly, so QH₂P = QH₃P = 0 throughout the full system, with P projecting onto the code and Q onto its complement. Their logical generators satisfy Lie(iA, iB) ≅ u(2) ⊕ u(1), allowing arbitrary coherent qubit gates while preserving the code at every intermediate time. The manuscript supplies exact certificates and verification procedures, together with operator-norm and calibration-error bounds that make the engineering requirements explicit. The practical opportunity is to remove intermediate code-check measurements from this control scheme; demonstrating a hardware advantage requires calibrated simultaneous interactions, including the diagonal channel, and measured gate fidelity and latency.
Two finite-order quantum controls may each be compatible with a qubit while their joint algebra is not. We study a cyclic control Q and an independent involution R with target algebra C_n x C_2. For even n >= 4, no faithful projective qubit realization exists: in PU(2) ≃ SO(3), a nontrivial involution commuting with an order-n rotation collapses to Q^(n/2). An explicit qutrit construction realizes the full algebra, so the minimum projective dimension is three; n = 4 is the first obstructed even member and has an eight-ray regular orbit. The representation-theory obstruction itself lies in a mature mathematical area. The operational contribution developed here is a calibrated qubit-consistency test: for exact quarter-turn and half-turn targets, the commutator angle gives a tight state-independent fidelity bound and a tight optimized binary-contrast cap. We derive a sufficient angular-error version and extend the test to general CPTP gate deviations through diamond distance. The explicit qutrit construction reaches the algebraic maximum binary contrast of one using two control sequences and a common binary measurement. The result is a calibrated falsification test for a qubit control model, not a device-independent dimension witness, and it remains entirely within standard finite-dimensional quantum mechanics.
Exchange-only quantum computing benefits from high fidelity and straightforward control afforded by the exchange interaction. However, independently controllable qubits must be encoded into subsystems of at least three electron spins, restricting computation to only a fraction of the available spin Hilbert space. The remaining states are treated as leakage and used only transiently in gate sequences. By contrast, a quantitative, system-wide measure of exchange-only performance can exploit the full available Hilbert space, including states conventionally treated as leakage. Here, we apply this approach to arrays of up to 48 electron spins, accessing a total-spin-zero Hilbert space with dimension exceeding 240. Measurements of out-of-time-order correlators (OTOCs) reveal rich scrambling dynamics and demonstrate access to regimes relevant to quantum computational advantage. Using generalized forms of cross-entropy and mirror randomized benchmarking, we also assess the aggregate performance of the full exchange-only system per fundamental two-body interaction: the two-spin exchange. We obtain an effective system-level error of 3×10−4 per exchange, incorporating the complete experimental control sequence and all associated noise sources. This value is up to an order of magnitude lower than those reported from comparable benchmarks on other platforms at the time of writing.
Measurement-based quantum computing (MBQC) promises an alternative approach to quantum computation that has natural compatibility with error-correction codes at the cost of a polynomial increase in physical qubits. MBQC implementations have previously focused on photonic systems where two-qubit gates are difficult. On the other hand, semiconductor spin qubit systems offer fast two-qubit gates via the exchange interaction. To explore the benefits of MBQC on spin systems, two hybrid measurement-exchange schemes for full qubit control with semiconductor double-quantum-dot spin qubits are considered. Protocol 1 fully realizes the cluster-state approach to MBQC but requires singlet-triplet qubits in a magnetic field gradient. Protocol 2 implements a direct measurement-based replacement for more traditional gate-based encoded operations, without need for magnetic field gradients. Both protocols demonstrate full single- and two-qubit control through a combination of interqubit and intraqubit exchange and measurement onto the singlet-triplet basis. We show that both schemes suppress individual qubit spin-state leakage errors and offer fast gate times, up to known phase and Pauli corrections.
Joint measurements of two-Pauli observables are powerful tools for both the control and protection of quantum information. By following a simple recipe for measurement choices, single- and two- qubit rotations using two-Pauli parity and single qubit measurements are guaranteed to be unitary while requiring only a single ancilla qubit. This language for measurement based quantum computing is shown to be directly applicable to encoded double quantum dot singlet-triplet spin qubits, by measuring spin-parity between dots from neighboring qubits. Along with exchange interaction, a complete, leakage free, measurement based gate set can be shown, up to a known Pauli correction. Both theoretically exact spin-parity measurements and experimentally demonstrated asymmetric spin-parity measurements are shown to be viable for achieving the proposed measurement based scheme, provided some extra leakage mitigating measurement steps. This method of spin qubit control offers a leakage suppressed, low resource overhead implementation of a measurement-based control that is viable on current spin qubit processor devices.
Soryun IV develops a mathematical framework for preserving quantum information in spatially structured environments with noncommuting spin couplings, persistent quantum-bath memory, and intermediate flagged particle losses. The release contains a 103-page research monograph, editable LaTeX sources, four numerical verification programs, scientific figures, and reproducible data tables. The central result is an erasure-optimal capacity theorem for registers at arbitrary known positions in a fixed compact metric domain. Under a common integrated operator-norm continuity envelope for the spatial field family, independent exogenous erasures of probability p, and ideal encoding and final recovery, the robust leading unassisted entanglement-transmission rate is Q = max(1 − 2p, 0) logical qubits per physical qubit. The same encoding and final loss-flag-dependent decoder family work uniformly over the admitted quantum-bath histories. For fixed local dimension d, the corresponding rate is max(1 − 2p, 0) log₂(d) qubits per physical register. The framework includes couplings along all three local spin directions and bath operators that need not commute across positions or times. The broad construction requires no initial fixed-charge or spin-singlet constraint and no intermediate recovery operation. Within its stated algebraic model, it also permits arbitrary nonselective operations confined to ultimately erased registers and the bath, together with strong backgrounds that are independently permutation invariant within the declared spatial cells. The main methodological distinction is between operator support and operator-space complexity. Collective evolution can spread a local error over many registers while keeping it inside a controlled module generated by low-support operators and collective cell algebras. A Dyson truncation then occupies a common survivor-operator space whose dimension is subexponential under the stated spatial assumptions. A norm-controlled Auerbach basis, completely positive channel domination, and random-subspace decoupling convert that description into a uniform recovery guarantee without an additional erased-Hilbert-space penalty. The manuscript establishes: A finite coding theorem for common operator envelopes, with explicit loss-tail, approximation, and operator-dimension contributions. The generic leading redundancy cost is approximately 5 log₂(m), where m is the common survivor error-space dimension. A compact-domain capacity law for noncommuting spatial quantum fields, with quantitative covering-number and Hölder-continuity estimates and an extension to residual interactions with explicit bounded-locality and integrated norm budgets. Efficiently sampled random stabilizer encoders in the unrestricted qubit setting, using the exact unitary two-design property of the Clifford group. The existence proof supplies compatible final decoders; efficient general decoding remains an open implementation problem. Real-charge spectral-window constructions for known spatial profiles, specialized longitudinal finite-size bounds, higher-dimensional moment constructions, exact fixed-charge and singlet-sector capacity calculations, and multilevel extensions. Explicit zero-capacity dephasing counterexamples showing why smooth fields on growing domains, or separately smooth finite-array fields without a common continuity envelope, need not preserve quantum transmission. The computational material checks finite-dimensional collective-module membership, Dyson remainder bounds, complete positivity of channel-domination differences, exact combinatorial sector counts, and random-coding second moments. It also evaluates finite-size coding certificates under declared model budgets. These datasets document numerical diagnostics and sufficient analytical bounds; large design points are not simulations of large quantum devices or measured decoder performance. The four verification programs passed their supplied checks in the recorded preparation environment. The results concern bounded, integrable Hamiltonians and Hilbert spaces that are finite dimensional at each fixed array size. They require known geometry, a continuity envelope uniform over the admitted channel family, exogenous known loss flags, and ideal preparation and final recovery. Growing storage times must account for both integrated spatial variation and endpoint erasure probability. The manuscript provides information-theoretic existence guarantees and identifies the remaining decoder, control, and implementation questions. Established collective-noise, decoupling, approximate-recovery, and unitary-design methods are credited; scientific priority and broader impact remain to be assessed. Creative inspiration included Suoming from Wuthering Waves.
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and d times that infidelity at leading order, for logical dimension d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is βr=E[sin2r(θ/2)], with θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(βr) exactly when generator powers through degree r−1 are correctable. When the first uncorrectable power has degree m<r, the optimal loss scales as βrm/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.