Complexity of Fermionic Dissipative Interactions and Applications to Quantum Computing

Interactions between particles are usually a resource for quantum computing, making quantum many-body systems intractable by any known classical algorithm. In contrast, noise is typically considered as being inimical to quantum many-body correlations, ultimately leading the system to a classically tractable state. This work shows that noise represented by two-body processes, such as pair loss, plays the same role as many-body interactions and makes otherwise classically simulable systems universal for quantum computing. We analyze such processes in detail and establish a complexity transition between simulable and nonsimulable systems as a function of a tuning parameter. We determine important classes of simulable and nonsimulable two-body dissipation. Finally, we show how using resonant dissipation in cold atoms can enhance the performance of two-qubit gates.

Popular Summary

Decoherence is a major obstacle in creating scalable quantum computers, as it generally destroys quantum correlations responsible for their advantage over classical computers. However, does it mean that any dissipation leading to decoherence is inherently harmful? As we show, some dissipative processes always lead a system to an essentially classical state. However, other dissipative processes can be useful for quantum computation. In fact, they can be so useful that they can replace two-qubit gates, as we illustrate in a blueprint for dissipation-based quantum computing.

To isolate the effect of dissipation, we study its effect on the computational complexity of simulating noninteracting fermions. We measure computational complexity by whether it is easy or hard to simulate a system on a classical computer. It is known that simulating the dynamics of free fermions is easy, and we extend this result by incorporating some two-body dissipative processes. We also show there exist types of two-body dissipation that convert noninteracting fermion dynamics into universal quantum dynamics, thereby proving that these types of dissipation processes are hard to simulate classically. Notably, the complexity of such dissipative processes is not affected by arbitrary small perturbations.

The existence of easy and hard classes potentially points to a new type of phase transition between classically easy and hard dynamics in many-body physics. Moreover, our analysis shows that it is possible to build a quantum computer based on dissipation, which is an exciting experimental prospect.

Figure 1

Classical simulability. We look for the existence of an efficient algorithm running on a classical computer and producing (sampling) the many-body configurations with the probability distribution close to the physical system after measurement in some basis. We show that, for fermionic systems with Hamiltonian $H(t)$ and with dissipation described by quadratic Lindblad jump operators $A_k(t)$, such an algorithm exists for at least a restricted number of problems, while the worst-case scenario requires a quantum computer in order to be solved efficiently. The three optical lattices illustrate the state of the system at initial, intermediate, and final times.

Figure 2

Complexity phase diagram for a fermionic system with simultaneous pair losses and gain. The plot illustrates the connection between complexity of simulation and the hypothetical dissipative control-$Z$ gate error $ϵ+{ϵ}^{\prime }$ (both axes use log scale). When the error is smaller than the best-known two-qubit error-correction threshold $p_0$, the worst-case system dynamics is equivalent to that of a fault-tolerant quantum computer (blue shaded regions) and, according to existing complexity conjectures, is classically computationally hard to simulate. In contrast, when the rates of gain and loss are exactly equal, the problem belongs to EC1 (vertical red line) with the effective classical algorithm provided in the text. The result for the unshaded region remains inconclusive. The dashed line represents qualitative extrapolation.

Figure 3

Magnetic Feshbach resonance for ${}^{40}\mathrm{K}$ atoms. (a) The hyperfine-Zeeman energy levels $E/h$ (GHz) of the $f=9/2$ and $7/2$ manifolds versus magnetic field $B$ (Gauss) [note also that we use G (Gauss) as the magnetic field unit in this paper because of its near-universal usage among groups working in this field, 1 $G={10}^{-4}$ T], labeled 1, 2, $\dots$ in order of increasing energy. The levels labeled $1,2,\dots$ have spin projections $m_f=-9/2,-7/2,\dots$, respectively. The spin and Zeeman coupling parameters are taken from Ref. [84]. (b) Scattering lengths of two ${}^{40}\mathrm{K}$ atoms. The solid and dashed lines represent $-\mathrm{\Im }({\tilde{a}}_0)\propto K_2$ and $\mathrm{\Re }({\tilde{a}}_0)$, respectively; blue and red represent the $1+4$ and $1+5$ channels, respectively. (c) Magnetic Feshbach resonance for the $1+4$ channel. The shaded regions depict magnetic fields where the elastic interaction between two atoms is smaller than the pair-loss rate, marking the regime of dissipative fermionic dynamics. All lengths are provided in Bohr-radius units $r_B=5.29\times {10}^{-9}$ cm, and the collision energy is $E/k_B=1\mu \mathrm{K}$, where $k_B$ is the Boltzmann constant.

Figure 4

Dissipative gate utilizing cold atoms. (a) Two-qubit logical states encoded using a pair of atoms in distinct states occupying four neighboring sites. (b) The upper bound for the total dissipative gate error as a function of magnetic field for ${}^{40}\mathrm{K}$, given the background error rate ${\mathrm{\Gamma }}^{\prime }={10}^{-2}{\mathrm{s}}^{-1}$. The shaded areas describe the weak interactions regime shown in Fig. 3.

Figure 5

Performance of Wick’s theorem. The plot shows the dependence of error in Eq. (D5) as a function of time for three processes, involving quadratic jump operators from EC1 (blue, circles), unitary jump operators from EC2 (orange, squares), and linear jump operators from EC3 (green, triangles) as discussed in the text of Appendix pp4. The systems size is $L=4$. The parameters are chosen as $\beta J=\beta {\mathrm{\Gamma }}_q=\beta {\mathrm{\Gamma }}_s=\beta J_u=\beta {\mathrm{\Gamma }}_u=1$, $\beta {\mathrm{\Gamma }}_s^{\prime }=2$. The plot demonstrates that jump operators from EC1 and EC2 violate Wick’s theorem, in contrast to the linear jump operators.