arXiv subjects

Henry Shackleton

Publications and source records attributed to Henry Shackleton.

4 recordsLinked to original sources

arXiv2025

The source of hardware-tailored codes and coding phases

A central challenge in quantum error correction is identifying powerful quantum codes tailored to specific hardware and determining their error thresholds above which quantum information is unprotected. This problem is hard because we cannot determine the noise models for our devices. Inspired by the quantum capacity theorem, we seek an optimal quantum source of information, namely the density matrix that degrades minimally when passed through a noisy channel. We explore this idea with the Open Random Unitary Model (ORUM), a simplified model of a $N$-qubit quantum computer with competing depolarizing and dephasing channels as a stand-in for unitary gates and measurements. Through numerical optimization, we find that the ORUM hosts three discrete regimes, three "phases", the "maximally mixed source" phase, a "$\mathbb{Z}_2$ source" phase (where ORUM's $U(1)$ gauge symmetry is broken down to $\mathbb{Z}_2$), and a no-coding phase where all information is lost. These phases exhibit first-order transitions among themselves and converge at a novel zero-capacity multicritical point. These results show a remarkable similarity between the quantum capacity theorem and Jaynes' maximum entropy principle of statistical mechanics. Using the $\mathbb{Z}_2$ source, we build two codes, a classical cat code capable of correcting all the dephasing errors and a concatenated cat code capable of correcting all errors up to a distance $d=\text{min}(m,N)$ and reduces to Shor's 9-qubit code for $m=N=3$. Neither classical nor quantum code survives near the vicinity of the zero-capacity multicritical point in the source phase diagram. Applying our approach to current noisy devices could provide a systematic method for constructing quantum codes for robust computation and communication.

quant-ph↗

arXiv2024

Deconfined quantum criticality of nodal $d$-wave superconductivity, N\'eel order, and charge order on the square lattice at half-filling

We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the $\pi$-flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background $\pi$-flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal $d$-wave superconductor, and states with N\'eel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases which are very likely described by 2+1 dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and N\'eel order, and a conventional $d$-wave superconductor with gapless Bogoliubov quasiparticles at 4 nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of N\'eel, valence bond solid (Kekul\'e), and Dirac semi-metal phases.

cond-mat.str-el↗

arXiv2023

A model of $d$-wave superconductivity, antiferromagnetism, and charge order on the square lattice

Early studies proposed a connection between cuprate superconductivity and fractionalized spin liquid states. But the low temperature phase diagram is dominated by states without fractionalization, with a competition between superconductivity and charge-ordered states which break translational symmetry. Our theory uncovers novel features associated with a particular spin-liquid presumed to underlie the pseudogap metal, and shows that it has multiple nearly-degenerate instabilities to confinement of fractionalized excitations, leading to antiferromagnetism, $d$-wave superconductivity, and/or charge order. Our theory provides routes to resolving a number of open puzzles on the cuprate phase diagram. The spin liquid is described by a SU(2) gauge theory of $N_f=2$ massless fundamental Dirac fermions, has an emergent SO(5)$_f$ global symmetry, and is presumed to confine at low energies to the N\'eel state. At non-zero doping (or smaller Hubbard repulsion at half-filling) we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges moving in $\pi$ flux. At half-filling, the low energy Higgs sector has $N_b=2$ relativistic bosons with a possible emergent SO(5)$_b$ global symmetry describing rotations between a $d$-wave superconductor, period-2 charge stripes, and the time-reversal breaking `$d$-density wave' state. We propose a deconfined quantum critical point between a confining state which breaks SO(5)$_f$ and a confining state which breaks SO(5)$_b$. The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between N\'eel order and $d$-wave superconductivity. A similar theory applies at non-zero doping and large $U$, with longer-range couplings of the chargons leading to charge order with longer periods.

cond-mat.str-el↗

arXiv2017

Pinning of Diffusional Instabilities by Non-Uniform Curvature

Turing patterns emerge from a spatially uniform state following a linear instability driven by diffusion. Features of the eventual pattern (stabilized by non-linearities) are already present in the initial unstable modes. On a uniform flat surface or perfect sphere, the unstable modes and final patterns are degenerate, reflecting translational/rotational symmetry. This symmetry can be broken, e.g. by a bump on a flat substrate or by deforming a sphere. As the diffusion operator on a two dimensional manifold depends on the underlying curvature, the degeneracy of the initial unstable mode is similarly reduced. Different shapes can pin different modes. We adapt methods of conformal mapping and perturbation theory to analytically examine how bumps and ripples entrain modes of the diffusion operator on cylinders and spheres. We confirm these results numerically, and provide closed form expressions that describe how non-uniformities in curvature pin diffusion-driven instabilities and the resulting patterns.

cond-mat.soft↗