Enhancing quantum computation via superposition of quantum gates

Overcoming the influence of noise and imperfections in quantum devices is one of the main challenges for viable quantum applications. In this article, we present different protocols, which we denote as “superposed quantum error mitigation,” that enhance the fidelity of single gates or entire computations by performing them in coherent superposition. Our results demonstrate that via our methods, significant noise suppression can be achieved for most kinds of decoherence and standard experimental parameter regimes. Our protocols can be either deterministic, such that the outcome is never postselected, or probabilistic, in which case the resulting state must be discarded unless a well-specified condition is met. By using sufficiently many resources and working under broad assumptions, our methods can yield the desired output state with unit fidelity. Finally, we analyze our approach for gate-based, measurement-based, and interferometric-based models, demonstrating the applicability in all cases and investigating the fundamental mechanisms they rely upon.

Figure 1

Illustration of our SQEM method applied to an arbitrary computation $U$ acting on $m$ input qubits. (a) The fidelity $F$ is calculated with respect to the output of the perfect computation. In the noiseless case, the absence of error guarantees unit fidelity $F=1$. When considering realistic settings, noise acting on the system lowers the fidelity $F<1$. In (b) and (c) we schematically represent the underlying idea behind our protocols. By creating a superposition of two (or more) identical computations, it is possible to enhance the fidelity of the computation.

Figure 2

Schematic representation of Protocol lg1. Vertical dashed lines, marked with numbers from one to six, identify the system's state after the corresponding steps of the protocol, as explained in the main text. We explicitly include the initial $|{\psi }_{\mathrm{in}}{\rangle }_a$ and auxiliary $|{\varphi }_0{\rangle }_{b_i}$ ($i=1,\cdots ,d-1$) states, as well as the noisy computations ${\mathcal{E}}_U$. The multiqubit gates between lines 1 and 2, as well as 3 and 4, are CSWAP operations defined in Eq. (20).

Figure 3

Schematic representation of the evolution of correlations between the systems and environments during the process.

Figure 4

Example of three concatenations of the nested approach that allows one to effectively implement our protocol with a superposition of $n$ channels by using only the basic CSWAP operations [Eq. (20)] for $d=2$.

Figure 5

Infidelity ratio $\mathcal{R}$ [see Eq. (41)] of the probabilistic protocol for a $U=T$ gate with depolarizing noise, $d=2$ and $p_0=0.97$ [see Eq. (9b)]. In (a), the auxiliary state is set to $|{\varphi }_0\rangle =cos(\theta /2)|0\rangle +sin(\theta /2)e^{i\varphi }|1\rangle$, while the measurement basis is the $X$ basis. In (b), the auxiliary state is $|{\varphi }_0\rangle =\left(\right|0\rangle +e^{i\theta }|1\rangle )/\sqrt{2}$ and the measurement basis contains the element $\left(\right|0\rangle +e^{i\varphi }|1\rangle )/\sqrt{2}$. In both panels, the chosen state $|{\varphi }_{\mathrm{f}}\rangle$ for the postselection at step 5 of the protocol is the one characterized by the largest probability.

Figure 6

Performance of the probabilistic (first column), quasideterministic (second column), and completely deterministic (third column) protocols applied to a T gate (first row, $m=1$) and a cnot gate (second row, $m=2$). For the T (cnot) gate we consider dephasing (depolarizing) noise, with single-qubit no-error probability $p_0$ (see Sec. 2b1). Colors are used to indicate different values of $d$, as shown in the legend, and we set $({\omega }_1,{\omega }_2)=(1,1)$. For the quasideterministic and completely deterministic protocols, only single-qubit Clifford operations are considered as correcting unitaries (see Sec. 4c2).

Figure 7

Performance of the nested protocol applied to a cnot ($m=2$) with depolarizing noise. In (a) we vary the single-qubit no-error probability $p_0$ (see Sec. 2b1), while in (b) we vary the number of iterations $n$ and the associated total number of auxiliary states (equivalent branches) employed $d_{\mathrm{tot}}={\sum }_{k=0}^{n-1}{d}_k$. Dark blue dots are obtained with ${\omega }_1=1$, and represent the upper bound on the ratio $\mathcal{R}$ that can be achieved by our protocols (hence the shadowed area). Light blue dots are derived for $|{\varphi }_k\rangle =|++\rangle$ for all $k=0,\cdots ,n-1$, while blue dots are found from the auxiliary states in Eq. (40). Vertical lines indicate the values of $p_0=0.9$ and $n=12$ (i.e., $d_{\mathrm{tot}}=4096$) that are employed in the other panel. We set ${\omega }_2=1$ for clarity, and numerical results are obtained by simulating Eqs. (39).

Figure 8

Performance of the quasideterministic protocol ($d=2$) for circuits U of different depth. The number of layers $N_{\mathrm{L}}$ of cnot and T gates (corresponding to the circuit depth) is plotted against $\mathcal{R}$. Success probability is given in the inset. A single layer comprises a cnot followed by two T gates applied to each of the $m=2$ qubits. The auxiliary is chosen such that ${\omega }_2=1$ and the noise is modeled by depolarizing channels after each gate, with $p_0=1-3\times {10}^{-4}$. As reported in the legend, colored dotted lines are associated with different relative error probabilities, i.e., $p_{\mathrm{relative}}=(1-p_{\mathrm{CSWAP}})/(1-p_0)$ characterizing the depolarizing channels acting after each of the two Fredkin gates that a CSWAP is decomposed into.

Figure 9

Schematic representation of the SQEM process for an MB-QC implementation, which enhances the fidelity of any noisy computation ${\mathcal{E}}_{\mathrm{U}}$. A control register of dimension $d$ generates the superposition by swapping in a controlled way the $m$-qubit input state with $dm$-qubit auxiliaries. All the operations can be performed in a measurement-based fashion, where noise arises from the imperfect preparation of the resource state.

Figure 10

Performance of the Protocol lg2 for mitigating the noise of a T gate and a cnot gate in an MB setting, whose imperfect implementation is modeled by depolarizing noise with no-error probability $p_0$ acting on every qubit of the resource state. In this case, the auxiliary state, measurement bases, and correcting operations are all chosen from the Clifford group, such that ${\omega }_2<1$ for the T gate, ${\omega }_2=1$ for the cnot gate, and ${\omega }_1<1$.

Figure 11

Advantage ratio, Eq. (41), of MB-QC with noisy CSWAP gates, where each CSWAP operation is implemented in an MB fashion, and each qubit involved is subjected to the same depolarizing noise (with no-error probability $p_0$) as the rest of the resource qubits (i.e., the ones that carry out the computation). As in the GB case, each of the $N_{\mathrm{L}}$ layers comprises a cnot followed by two T gates applied to each of the $m=2$ qubits.

Figure 12

Illustration of the strategy for enhancing the fidelity of a noisy computation in an interferometriclike circuit model. The noisy computation is identically applied in all branches of the superposition and no dedicated auxiliary system is needed. The control register can be encoded in some extra degree of freedom of the input qubit(s). System-vacuum correlations (see text) allow us to dilute the effect of the noise and lead to computational fidelity advantage.

Figure 13

Performance of IB SQEM protocol applied to a T gate and a cnot gate, affected by depolarizing and dephasing noises, respectively. The performance in our coherent protocol is based on the stochastic magnetic field models (54) and (60) with each field component following the Gaussian distribution (55); the vacuum interference operators of these models are given by Eqs. (59) and (62), respectively.

Figure 14

(a) Operational representation of steps for the entanglement-based teleportation protocol. Once these steps are completed, final measurements on the remaining noisy Bell state $a_2{b}_2$ and on the control register are performed. (b) Schematic representation of the joint state of the system after each step in (a).

Figure 15

Graph states and measurement patterns used for the numerical simulations of the CSWAP gate in the measurement-based setting. Black qubits indicate measurements in the Pauli $X$ basis, while green qubits indicate adaptive measurements (see text).