Quantum Zeno effect in correlated qubits

Near-term quantum hardware promises to achieve quantum supremacy. From a quantum dynamical point of view, however, it is not unambiguously clear whether fundamental peculiarities of quantum physics permit any arbitrary speedups in real time. We show that an only recently unveiled property of the quantum Fisher information has profound implications for the rate of possible quantum information processing. To this end, we analyze an exemplary and pedagogical example for a quantum computer consisting of a computational qubit and a quantum memory. We find that frequent interaction between memory and device exhibit the quantum Zeno effect. In a second part, we show that the Zeno effect can be prevented by carefully designing the correlations and interaction between single elements of the quantum memory.

Figure 1

Sketch of simple quantum computer: A computational qubit $\mathcal{Q}$ interacts with the $k\mathrm{th}$ qubit of a quantum memory $\mathcal{M}$ consisting of $N$ qubits for an interval of length $\mathrm{\Delta }t$. The strength of interaction between $\mathcal{Q}$ and $\mathcal{M}$ is denoted by $\omega$, and the strength of interaction between the $N$ qubits in $\mathcal{M}$ is given by $\mathrm{\Omega }$.

Figure 2

Quantum speed limit [Eq. (6)] for the first interaction interval, that is, for $t\le \mathrm{\Delta }t$. Observe the discontinuity at $(t,\nu )=(0,1)$.

Figure 3

Minimal time ${\tau }_{\min }$ to complete a qubit flip in $\mathcal{Q}$ as a function of $r=\mathrm{\Omega }/\omega$. The first vertical line (green) marks the optimal ratio $r_{\mathrm{opt}}\approx 1.620000$ that leads to the fastest computation ${\tau }_{min}^{\left(\mathrm{opt}\right)}\approx 1.807022$. The second vertical line (red) denotes a critical point $r_{\mathrm{crit}}\approx 2.087532$, where ${\tau }_{\min }$ discontinuously increases. Further, for $r\to \infty$ we see that ${\tau }_{\min }$ asymptotically approaches ${\tau }_{min}^{(r\to \infty )}\approx 3.332162$. The horizontal line is a reference line that represents the ideal case of a minimal ${\tau }_{\min }$ for $N=1$.

Figure 4

Quantum speed limit $v_{\mathrm{QSL}}$ of $\mathcal{Q}$ [Eq. (6)] (red, dashed line), and fidelity $p_1(r,t)=\left.〈1\right|{\rho }_t\left|1\right.〉$ [Eq. (4)] (blue, solid line) for various ratios of interaction strengths $r=\mathrm{\Omega }/\omega$ and total times of interaction $\tau$. (a) Small interaction ratio ($r=0.2, \tau ={\tau }_{min}$). (b) Optimal interaction ratio ($r=r_{\mathrm{opt}}\approx 1.62, \tau ={\tau }_{min}$). (c) Supercritical interaction ratio ($r=2.7>r_{\mathrm{crit}}, \tau ={\tau }_{min}$). (d) Supercritical interaction ratio $r=2.7, \tau$ given by the first local minimum of $p_1\left(\tau \right)$. In (a)–(c), the system qubit is successfully erased at the end of the cycle. However, since the ratio $r$ in (c) is above the critical value, the time it takes to erase is much longer than in (a) or (b). In (d), the fidelity achieves a local minimum at the end of the cycle, but the system qubit fails to be fully erased.

Figure 5

Fidelity $p_1(r,\tau )=\left.〈1\right|{\rho }_{\tau }\left|1\right.〉$ as a function of $\tau$ and $r$. The constant function $f=0.001$ (blue) illustrates the minimums of the fidelity. The first (green) and the second (red) vertical line denote the optimal ratio $r_{\mathrm{opt}}$ and the critical ratio $r_{\mathrm{crit}}$ respectively. The horizontal line is a reference line that represents the ideal case of a minimal ${\tau }_{\min }$ for $N=1$.