Scaling of decoherence for a system of uncoupled spin qubits

Significant experimental progresses in recent years have generated continued interest in quantum computation. A practical quantum computer would employ thousands if not millions of coherent qubits, and maintaining coherence in such a large system would be imperative for its utility. As an attempt at understanding the quantum coherence of multiple qubits, here we study decoherence of a multi-spin-qubit state under the influence of hyperfine interaction, and clearly demonstrate that the state structure is crucial to the scaling behavior of n-spin decoherence. Specifically, we find that coherence times of a multi-spin state at most scale with the number of qubits n as , while some states with higher symmetries have scale-free coherence with respect to n. Statistically, convergence to these scaling behavior is generally determined by the size of the Hilbert space m, which is usually much larger than n (up to an exponential function of n), so that convergence rate is very fast as we increase the number of qubits. Our results can be extended to other decoherence mechanisms, including in the presence of dynamical decoupling, which allow meaningful discussions on the scalability of spin-based quantum coherent technology.

splitting Ω much larger than the nuclear-spin-induced inhomogeneous broadening (see Fig. 1), so that spin relaxation is negligible. The dominant single-spin decoherence channel is pure dephasing due to the nuclear spins. We explore how this mechanism affects a many-spin-qubit state by systematically examining a large number of superposed states in various forms. Specifically, if the fidelity of an n-qubit state decays as exp[− γ (t)], we clarify how γ(t) depends on the qubit number n or the number of basis states m (which could be exponentially large as compared to n). Our results from this broad-ranged exploration indicate decoherence scaling behavior ranging from scale-free up to sublinear to n, making the scale-up of a spin-based quantum computer a tractable endeavor.

Electron-nuclear spin hyperfine interaction
We consider n uncoupled electron spins in a finite uniform magnetic field, each confined (in a quantum dot, nominally) and interacting with its own uncorrelated nuclear-spin bath through hyperfine interaction: where ω α j is the nuclear Zeeman splitting of the α-th nuclear spin in the j-th QD (from here on j will always be used to label the QDs and the corresponding electron spin qubits), and A ja is the hyperfine coupling strength. The number of nuclear spins coupled to the j-th electron spin, N j , is in the order of 10 5 to 10 6 in GaAs QDs, and ∼10 3 in natural Si QDs.
The total Hamiltonian (1) is a sum of n fully independent single-spin decoherence Hamiltonians. The evolution operator for these n qubit can thus be factored into a product of operators for individual qubits. We present a brief recap of single-spin decoherence 13,28 properties in Method, and focus here on the multi-spin-qubit decoherence problem. Recall that inhomogeneous broadening corresponds to stochastic phase diffusion of an electron spin due to longitudinal Overhauser field, and is characterized by the time scale ⁎ T 2 . On the other hand, the narrowed-state free induction decay is caused by fluctuations in the transverse Overhauser field, and is characterized by the time scale T 2 . These two time scales are statistically independent because of independence between longitudinal and transverse Overhauser fields, as presented in Method. These two pure dephasing channels follow the same scaling law, i.e., , where n is the number of spin qubits in the system. Thus we can focus on the scaling analysis of either of them. In the following we employ ( )/ ( )   T n T 1 2 2 to represent the result, which is applicable to both dephasing channels.

Results
Multi-spin decoherence. For an n-spin system in a finite uniform magnetic field, the full Hilbert space is divided into n+ 1 Zeeman subspaces, labeled by = − / S k n 2 z , = , , , , degenerate states (in the absence of nuclear field), which has k spins in the 1 ( ≡ − 1 1) state and − n k spins in the |1 state. The local random Overhauser fields break The spectrum splits into n + 1 Zeeman sub-levels. k refers to the number of spins that point down. Each electron spin is coupled to local nuclear spins through hyperfine interaction, which produces a local field in the order of ∆B, so that the energy level for each Zeeman manifold is broadened to a band with width ∆ n B.
Scientific RepoRts | 5:17013 | DOI: 10.1038/srep17013 this degeneracy and lead to a broadening of the manifold ∆ ∼ n B (see Fig. 1). In all the following calculations, we use spin product states = −  x l l l r n r n r r 1 1 as the bases. Here l j r refers to the electron spin orientation along the z-direction in the j-th QD for state x r , and takes the value of 1 or 1 for notational simplicity.
For a superposed state x containing more than one product state, decoherence emerges due to the non-stationary random phase differences among the m product states x r 's: The number of product states in x , m, is also the Hilbert space size of concern because spin relaxation is generally negligible in a finite field and is not considered in this study. We treat the Overhauser field (both longitudinal and transverse components) semiclassically, accurate to the second order in its magnitude. The notation − B l l l z n r n r r 1 1 represents a sum of Overhauser fields from every QD, and is defined in Method. As a measure of decoherence of x caused by the hyperfine interaction, we use fidelity ] , which can be simplified in the presence of dephasing as 1 is the Overhauser field difference experienced by the two n-spin product states (see Method). Specifically, ik is solely determined by the number of spins that are opposite in orientation between bases x i and x k . Therefore, the fidelity depends on the structure of the interested state, i.e., the constituents and their weight in the superposed state, and single-qubit decoherence is only one of several important ingredients in the multi-qubit decoherence problem.
Classification of multi-spin decoherence. With our understanding of single-spin decoherence, and with fidelity of the collective decoherence for a multi-spin state x defined, we are now in position to clarify multi-spin decoherence in various subspaces of the n-spin system.
Case A: single product state. The simplest multi-spin state is a single product state. The random Overhauser fields experienced by the spin qubits create a random but global phase (relative to when the nuclear reservoir is absent). This global phase does not lead to any decoherence, as there is no coherence (phase) information stored in any product state.
Case B: two product states, with m = 2 and k ≥ 1. The simplest multi-spin state that can undergo dephasing consists of two product states. Here we choose a particular class of , with one state being fully polarized = | ⊗ b 1 n , while the other being from the k-th subspace with k spins in 1 . The fidelity of this state is ( In this case, dephasing time is inversely proportional to the square root of the number of spins prepared as 1 in k . A special example here is the GHZ state, , where the square root of the number of spin qubits is from the quadratic time dependence in the exponent of  . The worst case scenario for an x containing two product states is when they have completely opposite spins.
Case C: n ≥ m ≥ 2, k = 1. We now consider an x that is a superposition of m product states from the manifold with one spin in 1 .
. This state is slightly more general than the well-known W state, with a random weight and phase for each basis state. The fidelity of Here the upper bound (∞ means no decoherence) is approached when a particular product state dominates over all others in weight: 2 1 , so that we go back to Case A. The lower bound for decoherence time is scale-free with respect to n, when the whole system acts like a giant spin− / . The overall decoherence is determined by the phase differences between every pair of states from the C n k basis states as well as the population distribution. Since = − C C n k n n k , we limit our discussion below to ≤ / k n 2 without loss of generality. The phase difference θ r r 1 2 between a particular pair of | 〉 x r 1 and | 〉 x r 2 can involve Overhauser fields in 2j QDs, where ≤ j k. In the extreme case of 2j = n, they have completely opposite spins. After a straightforward derivation via combinatorial mathematics, the fidelity for this state is found to be  (   As in Case D, we can generalize | 〉 x E to | 〉 ′ x E by randomizing the weight d r 2 's, ≤ ≤ r 1 2 n . In Fig. 3 we plot our numerical results as compared with the analytical expression from Eq. (6). The size of error bars in Fig. 3 for random states rapidly vanishes with increasing n. Similar to Case D, the inset shows that the standard deviation of ( )/ ( )   T n T 1 2 2 scales with the Hilbert space size m in the form − . m 1 2697 . Since here m increases exponentially with n, the rapid suppression of error bar size as we increase n is not surprising. Consequently, the decoherence time for an arbitrary state | 〉 ′ x E adheres to the sublinear power-law − / n 1 2 as soon as > n 2.

Discussion
We have explored the scaling behavior of decoherence of n uncoupled electron spin qubits by investigating the fidelity of 5 classes of representative superposed states x . Our results are summarized in Table 1, where k is the number of spins in 1 in a product state that makes up of x . Typically, the pure dephasing rates are not related to the sub-Hilbert-space size m. Instead, they are usually sublinear power-law functions of the qubit number n, with the exponent determined by the single-spin decoherence mechanism. Furthermore, if x is constrained in a single subspace with a fixed k, ( ) Fidelity is one specific way to represent the environmental decoherence effects on a multi-qubit state, with equally weighted contributions from all the off-diagonal density matrix elements. We choose it vs. n from randomly generated states over the whole Hilbert space of the n-spin system. The solid line is generated by Eq. (6), using the equal-superposition state | 〉 x E . Inset: standard deviation of ( )/ ( )   T n T 1 2 2 vs. Hilbert space size m = 2 n . For each n, The results are generated from 100 randomly selected states.  Table 1. A summary of decoherence times of n uncoupled electron spin qbits under the influence of hyperfine coupling with local nuclear baths.
partially because there is no consensus measure for multi-qubit entanglement. Still, fidelity does provide hints on the robustness of certain entangled states against pure dephasing considered in this study. It should be noted that the results for the often-studied multipartite states, GHZ states and W states (presented in Cases B and C, respectively) coincide with their entanglement behaviors. The entanglement of W states (fidelity undergoes scale-free decay with respect to n) outperforms that of GHZ states (fidelity decay rate is proportional to n ) in terms of their robustness 31 . The independence on n by the W states is generic, insensitive to the behavior of single-qubit decoherence. The scalings revealed in our case studies can be qualitatively understood by counting the number of different spin orientations in any pair of product states. Among m product states making up an arbitrary state x , a large fraction of pairs have ( ) O n electron spins oriented in the opposite direction. If we average over all possible states assuming ≈ / d m 1 r 2 , the fidelity given by Eq.
(2) could be estimated as The decoherence rates are insensitive to m because of normalization and our equal-population assumption. More specifically, in the k-th manifold, the scaling law is / k 1 because any pair of states here is different at most in ( ) O k spins. This scale-free behavior (with respect to n and m) is quite generic 26,27 , and not dependent on single-qubit decoherence.
Our study here could be straightforwardly extended to other single-qubit decoherence mechanisms. In general, if the single-spin decoherence function is given by , the index of every power-law (− / 1 2) in Table 1 should be changed to ν − / 1 . For decoherence due to Gaussian noise under dynamical decoupling 32 , the decay functions have ν = 4 for spin echo and ν = 6 for two-pulse Carr-Purcell-Meiboom-Gill sequence, so that the decoherence scaling factors for the n-spin system become − / n 1 4 and − / n 1 6 , respectively. For spin relaxation induced by electron-phonon interaction that produces a linear exponential decay characterized by T 1 , the sub-Hilbert space spanned by a multi-qubit state is usually not fixed. So that a comprehensive understanding of the decay scaling power-laws requires further studies. Nevertheless, certain coherence terms in the n-spin system will still follow − n 1 scaling, same as what our dephasing study indicates.
Generally, decoherence of any class of multi-qubit states is independent of the Hilbert space size m. Whether it is scale-free or scales as a polynomial of n depends on the state-structure, while the specific power-law depends on the single-qubit decoherence mechanism. On the other hand, the variability of decoherence for arbitrary states decreases polynomially with increasing m because we only consider dephasing.
In conclusion, we find that the structure of a multi-qubit state is a critical ingredient in determining its collective decoherence. While different from DFS 33 , the scale-free states help identify Hilbert subspaces that are more favorable in coherence preservation for spin-based qubits under the influence of local nuclear spin reservoirs.

Method
Single-Spin Decoherence. For a single electron spin coupled to the surrounding nuclear spins in a finite magnetic field, the nuclear reservoir causes pure dephasing via the effective Hamiltonian 13,28 where N is the number of nuclear spins, Ω is the electron Zeeman splitting, and A α is the hyperfine coupling strength. The sums over α and α′ here are over all the nuclear spins in the single quantum dot (QD). The dephasing dynamics has two contributions: H A is the longitudinal Overhauser field, while V is the second-order contribution from the transverse Overhauser field. In a finite field, normally the former dominates, generating a random effective magnetic field of ∆ ∼ B 1 to 5 mT 9 on a quantum-dot-confined electron spin in GaAs. This random field leads to a stochastic phase and accounts for the inhomogeneous broadening effect characterized by a free induction decay at the time scale of ( ) is in the order of 10 ns. If the effect of H A is suppressed, such as through nuclear spin pumping and polarization 10 , V, which is second order in the transverse Overhauser field, leads to the so-called narrowed-state free induction decay, by which the off-diagonal elements of the spin density matrix decay at the time scale of T n 2 FID . In the manuscript and here we will simplify the notation for T n 2 FID to ( ) T n 2 , where n indicates the number of spin qubits in consideration. For a single spin, n = 1, and the narrowed-state decoherence function is given by: