Holonomic quantum computing in symmetry-protected ground states of spin chains

Consider a chain of n spin-1 particles, each coupled to its nearest neighbors via the Heisenberg coupling with Hamiltonian

$$\begin{equation} H_{n}^{\rm open}=J\sum_{j=1}^{n-1}\skew4\vec{S}_j\,{\cdot}\,\skew4\vec{S}_{j+1}, \end{equation} \tag{ 1 }$$

for J > 0. This Hamiltonian, which is D₂-invariant and gapped, describes a Haldane phase that is distinct from the trivial phase in that local perturbations that respect the D₂ symmetry cannot connect these phases without closing the gap [6, 7]. The ground state of such open chains is nearly four-fold degenerate, with a splitting that decays exponentially in the chain length. In particular, the singlet (triplet) states are ground states for even (odd) length n chains but the splitting scales like O((−1)ⁿ e^−n/ξn^−1/2) where ξ ≈ 6.03 is the correlation length, whereas the gap to bulk spin-2 excitations is Δ ≈ 0.41J [29, 30].

The four ground states correspond to two fractionalized spin-$\frac{1}{2}$ degrees of freedom, one at each edge [19, 29, 31]. In order to model the state of one edge only, say the left, we 'terminate' the right end of the chain with an additional spin-$\frac{1}{2}$ particle (which may be fictitious), coupling the termination spin-$\frac{1}{2}$ to the boundary spin-1 with the Heisenberg interaction so that the total Hamiltonian is

$$\begin{equation} H_{n}^{\rm term}=J\sum_{j=1}^{n-1}\skew4\vec{S}_j\,{\cdot}\,\skew4\vec{S}_{j+1}+J\skew4\vec{S}_n\cdot\skew4\vec{s}_{n+1}. \end{equation} \tag{ 2 }$$

This effectively fixes boundary conditions on the right edge (in physics language), or purifies the right edge mode (in the language of quantum information theory), leaving a two-fold degenerate ground state⁹. Though presented here as merely a mathematical device, the extra spin-$\frac{1}{2}$ system could be realized as part of the system.

Several facts justify this model of the edge states. Firstly, at the AKLT point the description is exact, as ground states of the terminated Hamiltonian are also ground states of the unterminated Hamiltonian due to the frustration-free property of the valence-bond solid AKLT ground state [28]. Secondly, away from AKLT, numerical results show that for chains of modest length, measuring the termination spin of a H^term_n ground state results in a high-fidelity approximation to a H^open_n ground state. Specifically, fidelities at the Heisenberg point exceed 0.998 for chains of length up to 12, decreasing roughly linearly with chain length. Finally, direct numerical simulation of the gates and measurements presented below make clear that the scheme works well even for quite short chains: single qubit gate fidelities at the Heisenberg point exceed 0.999 even for chains of length as short as six.

The remaining fractionalized edge degree of freedom can be used to encode a single qubit, with logical Pauli operators $\bar {\sigma }^{\widehat{m}}=\Sigma ^{\widehat{m}}_n$ for the length-n chain defined as global rotations around the $\widehat{m}$-axis:

$$\begin{equation} {\rm i}\, \Sigma^{\widehat{m}}_n=\left(\bigotimes_{j=1}^n \exp({{\rm i}\,\pi S^{{\widehat{m}}}_j})\right)\otimes \exp\left({\rm i}\,\frac{\pi}{2}\sigma^{{\widehat{m}}}\right). \end{equation} \tag{ 3 }$$

As these operators generate a projective representation of the D₂ symmetry (the Pauli group), this qubit encoding is well-defined throughout the phase.

The encoded qubit can be manipulated by adiabatically weakening the boundary spin coupling and turning on a local term, as in the Hamiltonian

$$\begin{equation} H_{n}(t)=f(t)J(S^{\widehat{z}}_1)^2+g(t)J\skew4\vec{S}_1\,{\cdot}\,\skew4\vec{S}_{2}+H_{n-1}^{\rm term}, \end{equation} \tag{ 4 }$$

with monotonic f,g obeying f(0) = g(T) = 0 and f(T) = g(0) = 1. This squeezes the qubit into a slightly shorter chain, as the boundary spin is now in a product state with the remainder of the chain. Note that the time-dependent part of the Hamiltonian in equation (4) is D₂-invariant, and so preserves the degeneracy of the ground state.

To determine the effect of the single-qubit dynamics, we make use of two conserved quantities: $\Sigma _n^{\widehat{z}}$ and $\Sigma _n^{\widehat{x}}$. The former is clearly conserved; to see that the latter is, too, note that $[(S^{\widehat{z}})^2,\exp ({\mathrm {i}\,\pi S^{\widehat{x}}})]=0$ and in particular, $\exp (\mathrm {i}\,\pi S^{\widehat{x}}) | S^{\widehat{z}}=0 \rangle =- | S^{\widehat{z}}=0 \rangle$. Now imagine the qubit starts in a +1 eigenstate of $\Sigma _n^{\widehat{z}}$, i.e. the state

$$\begin{equation} \left | \psi(0) \right \rangle=\left | 0 \right \rangle_n\equiv\left | \Sigma_n^{\widehat{z}}=+1,H_{n}^{\rm term}=0 \right \rangle\,. \end{equation} \tag{ 5 }$$

After the adiabatic dynamics it becomes

$$\begin{equation} \left | \psi(T) \right \rangle=\left | \Sigma_n^{\widehat{z}}=+1,(S_1^{\widehat{z}})^2=0,H_{n-1}^{\rm term}=0 \right \rangle\,. \end{equation} \noindent \tag{ 6 }$$

But due to the product form of $\Sigma _n^{\widehat{z}}$, this is nothing other than the state $|S^{\widehat{z}}=0\rangle \otimes |\Sigma _{n-1}^{\widehat{z}}=+1, H_{n-1}^{\rm term}=0\rangle$, meaning $| \psi (T)\rangle =|S^{\widehat{z}}=0\rangle \otimes |{0}\rangle _{n-1}$, up to some unknown phase. By the same argumentation, |1〉_n becomes $|S^{\widehat{z}}=0\rangle \otimes |{1}\rangle _{n-1}$, up to a possibly different phase.

Up to an irrelevant global phase, the effect of the dynamics is a rotation of the qubit around the ${\widehat{z}}$-axis, the amount depending on the relative phase accumulated by the two states |0〉_n and |1〉_n. The relative phase is fixed by the other conserved quantity. Consider the time evolution of a logical qubit initialized in the +1 eigenstate of $\Sigma _n^{\widehat{x}}$, | + 〉_n. Because $\exp (\mathrm {i}\,\pi S^{\widehat{x}})|S^{\widehat{z}}=0\rangle =-|S^{\widehat{z}}=0\rangle$, the eigenvalue of $\Sigma _{n-1}^{\widehat{x}}$ in the final step must be −1, so | + 〉_n is transformed into $|S^{\widehat{z}}=0\rangle \otimes |-\rangle _{n-1}$. Therefore, the dynamics effects a π rotation of the qubit about the ${\widehat{z}}$ axis.

The dynamics can just as well be run in reverse, meaning we can perform any single qubit operation by first uncoupling and then recoupling the boundary spin. Aligning the local field to ${\widehat{m}}$ during the forward stage and to ${\widehat{m}}'$ during the reverse results in a qubit rotation of $2\cos ^{-1}({\widehat{m}}\cdot {\widehat{m}}')$ around the axis ${\widehat{m}}\times {\widehat{m}}'$. The local field rotation on the boundary spin should also be done adiabatically with respect to the local gap which we assume is ∼Δ; see figure 2.

Zoom In Zoom Out Reset image size

Two-qubit operations can be similarly realized by appropriately coupling the ends of two neighboring chains A and B while decoupling them from their respective chains. For a judicious choice of coupling, this results in a cphase gate followed by a joint π rotation about the $\widehat{x}$-axis. The Hamiltonian for this process, H^AB consists of the time-independent part H^A_n−1 + H^B_n−1 and a time-dependent part

$$\begin{equation} H^{AB}(t)=f(t)JW^{AB}+g(t)J(\skew4\vec{S}_1^A\,{\cdot}\,\skew4\vec{S}_{2}^A+\skew4\vec{S}_1^B\,{\cdot}\,\skew4\vec{S}_{2}^B), \end{equation} \tag{ 7 }$$

where the interaction term is given by

$$\begin{equation} W^{A\!B}=[(S_1^{\widehat{x}})^2-(S_1^{\widehat{y}})^2]^A\otimes [S_1^{\widehat{z}}]^B+[S_1^{\widehat{z}}]^A\otimes [(S_1^{\widehat{x}})^2-(S_1^{\widehat{y}})^2]^B. \end{equation} \tag{ 8 }$$

The ground state of this interaction term is

$$\begin{equation} \left | \xi \right \rangle^{AB}=\frac 12(-\left | xx \right \rangle+\left | xy \right \rangle+\left | yx \right \rangle+\left | yy \right \rangle)^{A\!B}, \end{equation} \tag{ 9 }$$

for |j〉 ≡ |S^j = 0〉.

Again the argument is based on various conserved quantities that arise from symmetries of the interaction, which are shown in table 1. The Haldane phase of two chains is not known to be protected by these symmetries, but numerical calculation on chains of moderate length confirms that the degeneracy is indeed maintained. This is implicitly shown in figure 3, which also depicts the gap to the excited states.

Zoom In Zoom Out Reset image size

Table 1. Symmetry operators of the interaction W^AB of equation (7), their eigenvalues in its ground state, and the corresponding conserved quantities for two chains under the associated dynamics. The first factor in the tensor product acts on system A, the second on B. The conserved quantities fix the action on the encoded qubits to be the cphase gate (followed by local rotations). The symmetry operators are all rotations, defined by $R^{\widehat{m}}=\exp (-\mathrm {i}\,\pi S^{\widehat{m}})$ and $\sqrt {R^{\widehat{m}}}= \exp (-\mathrm {i}\,\frac {\pi }{2} S^{\widehat{m}})$, with ${\widehat{u}}=\frac {1}{\sqrt {2}}({\widehat{x}}+{\widehat{y}})$ and ${\widehat{v}}=\frac {1}{\sqrt {2}}({\widehat{x}}-{\widehat{y}})$. Because the terms besides W^AB in the Hamiltonian are rotationally-invariant, applying these rotations to their entire respective chains leads to the listed conserved quantities.

W symmetry	Eigenvalue of |ξ〉	Conserved quantity
$R^{\widehat{z}}\otimes {\mathbbm 1}$	−1	$\Sigma ^{\widehat{z}}\otimes {\mathbbm 1}$
${\mathbbm 1} \otimes R_{\hat {z}}$	−1	${\mathbbm 1}\otimes \Sigma ^{\widehat{z}}$
$R^{\widehat{u}}\otimes R^{\widehat{u}}$	1	$\Sigma ^{\widehat{u}}\otimes \Sigma ^{\widehat{u}}$
$R^{\widehat{v}}\otimes R^{\widehat{v}}$	1	$\Sigma ^{\widehat{v}}\otimes \Sigma ^{\widehat{v}}$
$\sqrt {R^{\widehat{z}}}\otimes R^{\widehat{x}}$	−i	$\sqrt {\Sigma ^{\widehat{z}}}\otimes \Sigma ^{\widehat{x}}$

To see that this operation implements a two-qubit unitary gate, we group the set of conserved quantities according to the eigenvalue of |ξ〉 and start with the first two, corresponding to −1. The joint eigenstates are product encoded states, and it is immediately clear, using the same analysis as the single qubit case, that the action of the dynamics is given by an operator of the form

$$\begin{equation} U^{AB}=\left(\begin{array}{@{}llll@{}}0 & 0 & 0 & \alpha\\ 0 & 0 & \beta & 0\\ 0 & \gamma & 0 & 0 \\ \delta & 0 & 0 & 0\end{array}\right), \end{equation} \tag{ 10 }$$

where α,β,γ,δ are complex numbers of unit magnitude. Now we consider the second pair of conserved quantities and determine that their joint eigenstates are the (unnormalized) states |01〉 ± |10〉 and |00〉 ± i|11〉. Working out the action of U^AB on these states and using the overall +1 eigenvalue of |ξ〉 fixes δ = −α and γ = β. Finally, the last conserved quantity has a nondegenerate spectrum, and so does not require a partner to determine a basis for the encoded states from its eigenstates, which happens to be the canonical Bell states. Again applying U^AB and using the overall eigenvalue −i of |ξ〉 as before gives α = −β. Thus,

$$\begin{equation} U^{AB}=\left(\begin{array}{@{}llll@{}} 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right) \end{equation} \tag{ 11 }$$

up to an irrelevant overall phase, which is nothing other than $(\bar {\sigma }^{\hat {x}}\otimes \bar {\sigma }^{\hat {x}})$ cphase.

To initialize and measure the encoded quantum information, we may appeal to a method developed for a measurement-based precursor to the present scheme [12]. Here we again adiabatically turn off the boundary spin coupling, but now do not turn on any local field. Instead, after the coupling is off, the boundary spin is measured in the basis |S_z = m〉. A result m = 1 (m = −1) corresponds to a projection of the qubit onto |0〉 (|1〉), while m = 0 corresponds to a π rotation around the $\hat {z}$-axis. This can be understood as a consequence of addition of angular momentum; since the initial state has spin-$\frac{1}{2}$ and the dynamics is adiabatic and preserves rotational invariance, the output state also has spin-$\frac{1}{2}$. But it is now more naturally expressed as a combination of spin-1 and spin-$\frac{1}{2}$, and by using the Clebsch–Gordan coefficients for this case we can quickly deduce the effect of measuring the spin-1 system.

Once the system is in the ground state, encoded qubits may be initialized via this procedure by simply measuring until an |m| = 1 outcome is obtained. This produces either |0〉_n or |1〉_n and the computation can begin. The problem of initialization is therefore reduced to preparing the system in the ground state, which can be done either by preparing a ground state at some point in the phase convenient to do so, e.g. the AKLT point, and then slowly altering the Hamiltonian to any other point in the Haldane phase which is convenient to implement, e.g. the Heisenberg antiferromagnet or by actively cooling the state at a fixed point in the phase. For instance, Kraus et al [32] make use of the frustration-free property of the AKLT state to construct a Liouvillian map acting on neighboring pairs of spins and a local Markovian environment whose output converges to the AKLT ground subspace in a time which scales linearly with the chain length.

The procedure of [12] may also be used to read out the encoded qubit state after the computation is complete. This process is indeterministic due to the m = 0 outcome, which can be dealt with in one of two ways. Firstly, for outcome m = 0, the boundary spin can be recoupled (after turning on a local field S²_z) and the measurement attempted again. Secondly, controlled-not gates to several other encoded qubits could first be performed, and then the encoded qubits measured simultaneously, taking as the outcome the majority of nonzero results.

Because we encode one qubit into a many body chain there are several possible locations for error that must be accounted for. A complete summary of error mechanisms and the effect on encoded quantum information is displayed in table 2. The main effects are either to produce an error on the encoded information directly or to couple to states outside the qubit subspace which we denote leakage error.

Table 2. Error mechanisms and effects on encoded quantum information. Here p_L(ℓ) are the logical(leakage) error probabilities, h is the perturbation, either to the system Hamiltonian (Memory) or the gate Hamiltonian (Gate), and ∥·∥ is the operator norm. The Haldane gap is Δ and we assume the time to perform a gate is O(1/Δ). Relatively benign are the systematic errors can be corrected using composite pulse sequences of length k yielding an effective logical error p_L = O((∥h∥/Δ)^k) [33], and leakage errors which are correctable by cooling. Other noise mechanisms yielding logical errors must be handled using quantum error correction.

Error type	Effect
Memory
D2-invariant	Logically protected
Bulk	pL=0, $p_{\ell }\sim (\frac {||h||}{\Delta })^2$
Boundary	$p_{\mathrm {L}} \sim \frac {||h||}{\Delta }$
Gate
D2-invariant	Logically protected
Quenched	Systematic $p_{\mathrm {L}}\sim \frac {||h||}{\Delta }$
Stochastic	$p_{L}\sim \frac {||h||}{\Delta }$

Note that bulk and boundary perturbations of the system produce very different behavior, due to the fact that noise in the bulk produces local excitations, incapable of immediately causing logical errors as they cannot distinguish the logical states. Provided bulk excitations are cooled at a rate faster than the dispersion, they will not propagate to the boundaries to cause logical errors. In contrast, the boundary spin-1 particles are effectively free spin-$\frac{1}{2}$ degrees of freedom susceptible to local energy shifts which translate into logical errors, albeit with an energy cost.

To gain more insight into why the bulk errors are relatively benign, consider the valence-bond solid as a caricature of the Haldane phase ground state (which is exact at the AKLT point) [28]. A half-infinite chain has the following form in the Schwinger representation [38]:

$$\begin{equation} \left | \mathrm{vbs} \right \rangle=\left(\alpha a_0^{\dagger}+\beta b_0^{\dagger}\right)\prod_{j\geqslant 1}\left(a_j^{\dagger} b_{j+1}^{\dagger}-b_j^{\dagger} a_{j+1}^{\dagger}\right)\left | {\rm vac} \right \rangle, \end{equation} \tag{ 12 }$$

where a^†_j and b^†_j are harmonic oscillator creation operators for modes a and b at site j with the constraint a^†a + b^†b = 2 for every j, while α and β are complex coefficients for basis states of the edge mode. Single spin rotations mix the creation operators linearly, and in particular a $\hat {z}$-axis rotation by θ maps a^† to a^†e^iθ/2 and b^† to b^†e^−iθ/2. Letting C^†_j,j+1 = (a^†_jb^†_j+1 − b^†_ja^†_j+1), it is easy to work out that a rotation R_j of site j produces the transformation

$$\begin{equation} R_jC^{\dagger}_{j,j+1}R_j^{\dagger}=\cos\frac{\theta}{2}C^{\dagger}_{j,j+1}-{\rm i}\sin\frac{\theta}{2}\!\left(a^{\dagger}_jb^{\dagger}_{j+1}+b^{\dagger}_ja^{\dagger}_{j+1}\right). \end{equation} \tag{ 13 }$$

The second term produces an excited state from the vacuum, and thus rotation of a site in the bulk of the chain leaves the encoded qubit unaffected while producing a linear superposition of ground and excited states. Note that the weights in the superposition depend on the amount of rotation. However, since the superposition does not depend on the encoded information, it can therefore be corrected without damaging the qubit, e.g. by cooling. On the other hand, a rotation of the boundary spin produces a superposition between the original encoded qubit in the ground state and a rotated version in the excited state. Again, the amount of rotation determines the amplitude of the excited state, and so larger rotations cost more energy.

To estimate the leakage probability, note that the excited states are split from the ground states by the Haldane gap. For a local perturbation h acting in the bulk the amplitude of coupling to the excited states is ∼∥h∥T_gate and since the three step holonomic gate time is T_gate = 3/Δ_min = O(1/Δ), where Δ_min is the minimum gap during each step, then the probability to leak into the excited states is p_ℓ ∼ (∥h∥/Δ)². On the other hand, the edge modes are shifted in energy by a local perturbation without gap protection implying a logical error over a gate time scaling like p_L ∼ ∥h∥/Δ.

While the hardware protected quantum gates we have described reduce error rates due to environmental noise and control error, software based quantum error correction is needed to achieve fully fault tolerant quantum computation. In some ways our gate mechanism constrains the type of architecture and choice of QECC. Firstly, because we rely on adiabatically turning interactions on and off, our architecture is likely to allow only nearest neighbor interactions between qubits. It may be possible that a nonlocal coupling could be engineered using, for example, an optical mode in a fiber [39]; however, such interactions are typically weak and could prove difficult to wire in a scalable system. Secondly, since our qubits are degenerate by design, there is no bias toward any particular local Pauli errors. This is unlike the situation with many physical realizations of qubits which are nondegenerate, such as hyperfine split ground states of trapped ions or atoms, superconducting phase qubits, etc, which are inherently more resilient against bit flip errors in the energy basis versus phase errors. This being the case, it is reasonable to chose QECCs, and a concatenation method which is unbiased toward X or Z error. Some particular QECCs and architectures are well suited to this very situation [1].

One suitable code is the Bacon–Shor 9 qubit QECC embedded in a two-dimensional (2D) spatial architecture. It was shown in  [40] that 'padding' a 2D array with ancilla qubits in between data qubit provides sufficient room to perform fault tolerant swapping of information even when restricted to nearest neighbor interactions. Assuming an adversarial local error model and equal error rates for memory and gate errors, the threshold for fault-tolerant computing in that architecture is p_th = 1.3 × 10⁻⁵. This can be improved using other software style strategies such as concatenated dynamical decoupling pulses [2]. These thresholds are about a factor of 10 worse than a nonlocally corrected architecture. The Bacon–Shor code has the advantage of low latency (i.e. overall faster performance) relative to other CSS codes such as the 7 qubit QECC and does not require cat state verification of ancilla. Furthermore, this code also accommodates measurement-free QEC with only a small reduction in the threshold [41] which would obviate the need for fitting high efficiency detectors within the lattice of chains.

An illustration of an architecture using a 2D array of parallel chains of encoded qubits is shown in figure 4. Here the chains are depicted with real spin-$\frac{1}{2}$ boundary spins, which could however be removed with some minor changes to the protocol. As stated above, except at the AKLT point, the ground state manifold suffers a small degeneracy splitting which decreases exponentially with the chain length. The splitting translates into a small timing error in logical gates and opens up a zero energy leakage channel to the other edge mode. Both could be corrected with QEC, but another possibility is to use both edge modes as qubits and process them by manipulating both ends of the chains.

Zoom In Zoom Out Reset image size

Because each qubit is a collective degree of freedom of a spin chain, it might be thought that a local error model is not appropriate. However, all the chains are separated in space far enough to have negligible interactions between chains. Any correlated errors within a spin chain effectively act as either a leakage error when they occur in the bulk of the chain which can then propagate to the boundaries to create a logical error, or directly as a logical error at the boundary.

Leakage errors require special care during error correction because once such an error occurs in one chain, subsequent gates between chains that are even perfect with the logical subspace can propagate error in the leakage subspace. This has shown not to be disastrous, however, provided one can perform leakage reduction at the lowest level of concatenation (i.e. at the level of the physical chains) [42]. An appealing hardware strategy for leakage reduction is to bathe the spin chains in a cold environment that removes excitations without requiring active monitoring of the system. Such a procedure does not restore the quantum information, but rather reduces leakage errors to standard logical qubit errors which can be corrected using quantum error correction.