Entanglemons: Cross-platform protected qubits from entanglement

TL;DR

The paper addresses scalable fault-tolerant quantum computation by proposing entanglemon qubits that store information in an entanglement phase $\beta$ arising from a collective CP(3) degree of freedom. It develops two platform-agnostic models, $U(1)^{\beta}$ and $\mathbb{Z}_{2}^{\beta}$, and provides a Schwinger-boson geometric-quantization framework to analyze their spectra and noise-immunity properties. The authors demonstrate, with explicit analyses and sketches of hardware implementations in trapped ions, superconducting circuits, graphene/silicon quantum dots, and graphene skyrmions, how intrinsic protection against depolarization and/or dephasing can be achieved with modest resource requirements ($d$ as small as 2–4). This approach highlights a new route to intrinsic fault-tolerance by leveraging nonlinear geometry of higher-dimensional phase spaces and emergent weakly coupled degrees of freedom, potentially reducing error-correction overhead and enabling broader platform applicability.

Abstract

A crucial ingredient for scalable fault-tolerant quantum computing is the construction of logical qubits with low error rates and intrinsic noise protection. We propose a cross-platform construction for such hardware-level noise-protection in which the qubits are protected from depolarizing (relaxation) and dephasing errors induced by local noise. These logical qubits arise from the entanglement between two internal degrees of freedom, hence - entanglemons. Our construction is based on the emergence of collective degrees of freedom from a generalized coherent state construction, similar in spirit to spin coherent states, of a set of such internally entangled units. These degrees of freedom, for a finite number of units, parametrize the quantized version of complex projective space $\mathbb{C}$P(3). The noise protection of the entanglemon qubit is then a consequence of a weakly coupled emergent degree of freedom arising due to the non-linear geometry of complex projective space. We present two simple models for entanglemons which are platform agnostic, provide varying levels of protection and in which the qubit basis states are the two lowest energy states with a higher energy gap to other states. We end by commenting on how entanglemons could be realized in platforms ranging from superconducting circuits and trapped ion platforms to possibly also quantum Hall skyrmions in graphene and quantum dots in semiconductors. The inherent noise protection in our models combined with the platform agnosticism highlights the potential of encoding information in additional weakly coupled emergent degrees of freedom arising in non-linear geometrical spaces and curved phase spaces, thereby proposing a different route to achieve scalable fault-tolerance.

Figures (6)

• Figure 1: Pictorial description of the general principle behind the entanglemon construction. a) A pictorial representation of a standard minimum uncertainty coherent state of a harmonic oscillator with a peak around a well-defined value in phase space. b) The generalization of a coherent state to a collective state of many SU(2) spins. The phase space for the collective spin coherent state is $S^2$ and hence the spin coherent state can also be parametrized by the Bloch sphere, however, unlike the coherent state the spin coherent state does not peak at a fixed value. There is a "fuzziness" due to quantum fluctuations which vanishes in the large-S limit in which the spin coherent states reduce to the standard coherent states. c) Generalization of the spin coherent state construction to $\mathbb{C}$P(3) coherent states. The collective state of a group of $\mathbb{C}$P(3) objects has a six-dimensional phase-space corresponding to $\mathbb{C}$P(3). Similar to (b) there is an inherent fuzziness in the peak value which in the $d \rightarrow \infty$ limit corresponds to the standard coherent states (classical limit). Locally, the phase-space can be represented by three separate Bloch spheres via a Schmidt decomposition (see sec. \ref{['subsec_nontech']}) which gives rise to the notion of spin, pseudospin and entanglement Bloch spheres.
• Figure 2: Pictorial description of the two simple models for entanglemons --- models $U(1)^\beta$ and $\mathbb{Z}^\beta_2$ --- in terms of the three Bloch spheres, spin (violet), pseudospin (green) and entanglement (red), which parametrize the generalized $\mathbb{C}$P(3) coherent state (see Fig. \ref{['fig1']}). Both models involve entanglement between spin and pseudospin degrees of freedom where the qubit states are easy-axis states, i,e spin and pseudospin point along the z-axis. For model $U(1)^{\beta}$, the quantum states corresponding to the qubit are the orbits in the entanglement Bloch sphere corresponding to different values of $\alpha$. These states have a continuous $U(1)^{\beta}$ symmetry corrsponding to the transformation $\beta \rightarrow \beta + \theta$; $\theta \in [0,2\pi)$. These two states represent the two lowest states of an anharmonic oscillator spectrum which are well separated from the higher energy states (due to anharmonicity) and from other oscillator modes of the $\mathbb{C}$P(3) system (see sec III. 4. and Fig. \ref{['specfig']}). Such an encoding gives the qubit excellent protection noise-induced depolarization errors, however, there is no protection from dephasing (see sec \ref{['sub_noise_protection_dep']}). Model $\mathbb{Z}^\beta_2$ represents a different encoding for the entanglemon, where the continuous $U(1)^\beta$ of the previous model is broken down to a discrete $\mathbb{Z}_2^\beta$ symmetry. The two states comprising the basis for the logical qubit Hilbert space are then superpositions of states which lie on the same $\alpha$ orbit but at opposite $\beta$ points. Explicitly written, the qubit states are $\ket{0} = (\ket{\beta_0}+\ket{\beta_\pi})/\sqrt{2}$ and $\ket{1} = (\ket{\beta_0}-\ket{\beta_\pi})/\sqrt{2}$ and the qubit is protected from both depolarization and dephasing errors (see sec \ref{['subsec_full_protection']}).
• Figure 3: Geometry of Quantum Integrability - Tetrahedron of possible states with the two possible ground states for the easy-axis case marked $\text{CGS}_1$ and $\text{CGS}_2$. The entanglemon excitation about $\text{CGS}_1$ corresponds to moving along the top edge whereas the other oscillator modes correspond to moving along the faces. The former has an anharmonic spectrum associated to the compact variable $\beta$, whereas the latter oscillator modes have the standard harmonic oscillator spectrum. The excitation spectra in the neighbourhood of $CGS_1$ along the grey rod is computed analytically (semi-classical large-$d$ approximation) in sec. \ref{['sec_spectrum']}
• Figure 4: The low-lying spectrum as obtained in eq. (\ref{['spec']}) for $u_p = 2 \Delta$, $u_z = 0.7 \Delta$ and $d = 4$ and $n_2 - n_3 = 0$. (a)Blue-Entanglemon modes with anharmonic spectrum, Red and Green - Oscillator modes with standard harmonic spectrum. b) The lowest two modes of the anharmonic spectrum for the entanglemon qubit states $\ket{0}$ and $\ket{1}$ in model $U(1)^{\beta}$. The energy difference between these two states is unique allowing for selective operations. The minima of the curve for the spectrum is generically at a non-integer value (between 2 and 3 for these parameters), representing the Berry phase contribution from a cyclic $2\pi$ rotation of $\beta$ (see text for details.)
• Figure 5: Constructing entanglemon qubits from the hyperfine states of trapped ions such as Yb. The hyperfine states correspond to two entangled states between electronic and nuclear spin. The coupled array of such $\mathbb{C}$P(3) units is then realized by coupling to the motion modes of the ion array, which induces a "spin"-dependent (where spin up and down are the two hyperfine states) pairwise and all-to-all coupling resulting in a collective state of all ions in the $F = 0,m_F = 0$ state or all in the $F = 0,m_F = 1$ state. The minimal hardware requirement to realize the noise-immunity features of model $\mathbb{Z}^\beta_2$ is two ions, i.e $d = 2$. More ions can be coupled to increase noise protection.