Effect of correlated decay on fault-tolerant quantum computation

We analyze noise in the circuit model of quantum computers when the qubits are coupled to a common bosonic bath and discuss the possible failure of scalability of quantum computation. Specifically, we investigate correlated (super-radiant) decay between the qubit energy levels from a two- or three-dimensional array of qubits without imposing any restrictions on the size of the sample. We first show that regardless of how the spacing between the qubits compares with the emission wavelength, correlated decay produces errors outside the applicability of the threshold theorem. This is because the sum of the norms of the two-body interaction Hamiltonians (which can be viewed as the upper bound on the single-qubit error) that decoheres each qubit scales with the total number of qubits and is unbounded. We then discuss two related results: (1) We show that the actual error (instead of the upper bound) on each qubit scales with the number of qubits. As a result, in the limit of large number of qubits in the computer, $N\to \infty$, correlated decay causes each qubit in the computer to decohere in ever shorter time scales. (2) We find the complete eigenvalue spectrum of the exchange Hamiltonian that causes correlated decay in the same limit. We show that the spread of the eigenvalue distribution grows faster with $N$ compared to the spectrum of the unperturbed system Hamiltonian. As a result, as $N\to \infty$, quantum evolution becomes completely dominated by the noise due to correlated decay. These results argue that scalable quantum computing may not be possible in the circuit model in a two- or three- dimensional geometry when the qubits are coupled to a common bosonic bath.

Figure 1

We analyze an $N$-qubit quantum computer in (a) two- and (b) three-dimensional geometry with the qubits coupled to a common bosonic bath. For a single qubit present, the interaction with the bath causes an independent decay rate of $\mathrm{\Gamma }$ between the qubit levels. When the whole ensemble of qubits is present, correlated decay causes correlated errors to build up across the whole computer. Since we are primarily interested in the $N\to \infty$ limit, we do not impose restrictions on the size of the sample; i.e, the spacing between the qubits may be much larger than the radiation wavelength, $d\gg {\lambda }_a$. For concreteness, we focus on a square and a cube arrangement of atoms with regularly spaced qubits, but the results are not sensitive to the precise shape and arrangement structure of the array.

Figure 2

The distribution of the eigenvalues of the single-qubit error Hamiltonian ${\widehat{H}}^j={\sum }_k{\widehat{H}}^{jk}={\sum }_k{F}_{jk}{\widehat{\sigma }}_{+}^j{\widehat{\sigma }}_{-}^k+F_{kj}{\widehat{\sigma }}_{-}^j{\widehat{\sigma }}_{+}^k$ in the $N\to \infty$ limit. The distribution has a width $\sigma ={(N{E}^{\prime }\left[F^2\right]/2)}^{1/2}$ (the quantity $E^{\prime }\left[F^2\right]$ denotes the expected value of the square of the coupling constants for all pairings $jk$ with $j$ fixed), which scales as $\sigma \sim \sqrt{lnN}$ in a 2D and $\sigma \sim N^{1/6}$ in a 3D geometry, respectively.

Figure 3

The structure of the couplings in the coupled equations of Eqs. (13). Each ${\alpha }_q$ initially couples to ($\sim N/2$) ${\beta }_{q\ominus k}$ coefficients, each of which couples to ($\sim N/2$) ${\alpha }_{q\ominus k\oplus k^{\prime }}$ coefficients, and this structure continues until ${\alpha }_q$ couples to all the coefficients in a specific subspace of the Hilbert space. Because of this nested structure, the number of coefficients that ${\alpha }_q$ couples to grows exponentially.

Figure 4

The evolution of ${\alpha }_Q\left(t\right)$ as a function of time for the initial condition ${\alpha }_Q(t=0)=1$; ${\alpha }_q(t=0)=0$ for $q\ne Q$; ${\beta }_q(t=0)=0$. The initial probability amplitude decays in time scales given by $t\propto 1/\sigma$, where $\sigma ={(N{E}^{\prime }\left[F^2\right]/2)}^{1/2}$. As a result, the decay time scale gets ever shorter as the total number of qubits in the computer is increased: $t\sim 1/\sqrt{lnN}$ in 2D, $t\sim 1/N^{1/6}$ in 3D, respectively.

Figure 5

The time evolution of the coherence (off-diagonal density matrix element) for qubit $j$, ${\rho }_{01}^j\left(t\right)$, for eight different initial conditions in a 16-qubit quantum computer. (a) Uniform initial condition ${\alpha }_q(t=0)={\beta }_q(t=0)=1/\sqrt{2^N}$, (b) 8 nonzero amplitudes with equal amplitudes and phases, (c) only 2 nonzero amplitudes, (d) 200 nonzero amplitudes with 50 of them $\pi$ out of phase with the rest, (e) 50 nonzero amplitudes with 8 of them $\pi$ out of phase with the rest, and [(f)–(h)] all nonzero probability amplitudes with random amplitudes and phases (slight nonrandomness in the phases is introduced to produce a small initial value of the single-qubit coherence). For comparison, the square of the analytical solution of Fig. 4 is also plotted for each initial condition (red dashed line). The initial decay is well predicted by the analytical solution (which is valid in the $N\to \infty$ limit) even for this low number of qubits.

Figure 6

The absolute value of the expansion coefficients at $t=0$, $\sigma t=1$, $\sigma t=2$, $\sigma t=3$, and $\sigma t=4$ respectively. This simulation is performed for parameters that are identical to those used in the simulation of Fig. 5. Starting from the initial two nonzero probability amplitudes, the state of the computer evolves to cover almost all of the Hilbert space by the time $\sigma t\sim 4$.

Figure 7

The distribution of the eigenvalues of the total exchange Hamiltonian ${\widehat{H}}_{\mathrm{eff}}$ in the $N\to \infty$ limit. The eigenvalues are distributed according to the Laplace distribution, $f_{\mathrm{\Lambda }}\left(\lambda \right)=\frac{1}{2{\sigma }_{\mathrm{eff}}}\exp \left(-\frac{\left|\lambda \right|}{{\sigma }_{\mathrm{eff}}}\right)$ whose standard deviation is given by ${\sigma }_{\mathrm{eff}}=\frac{1}{2}NE{\left[F^2\right]}^{1/2}$. The width of the spectrum scales as ${\sigma }_{\mathrm{eff}}\sim N^{1/2}lnN$ in 2D and ${\sigma }_{\mathrm{eff}}\sim N^{2/3}$ in 3D respectively. Since in both geometries the spectrum spreads faster than the spectrum of the system Hamiltonian, $\frac{1}{2}{\omega }_a{\sum }_{j=1}^N{\widehat{\sigma }}_z^j$, the dynamics of the computer become completely dominated by the noise due to correlated decay in the $N\to \infty$ limit.

Figure 8

The probability density functions of the random variable $U\equiv 1/\left(k_a^2{r}_{ij}^2\right)$ for an array of atoms in two (dashed red line) and three (solid black line) dimensions. The functions are plotted for the case when the length of each side is $L=1$. For an arbitrary $L$, the horizontal axis of the plot is scaled by $1/L^2$ whereas the vertical axis is scaled by $L^2$.