Achieving metrological limits using ancilla-free quantum error-correcting codes

Quantum error correction (QEC) is theoretically capable of achieving the ultimate estimation limits in noisy quantum metrology. However, existing quantum error-correcting codes designed for noisy quantum metrology generally exploit entanglement between one probe and one noiseless ancilla of the same dimension, and the requirement of noiseless ancillas is one of the major obstacles to implementing the QEC metrological protocol in practice. Here we successfully lift this requirement by explicitly constructing two types of multiprobe quantum error-correcting codes, where the first one utilizes a negligible amount of ancillas and the second one is ancilla free. Specifically, we consider Hamiltonian estimation under Markovian noise and show that (i) when the Heisenberg limit (HL) is achievable our codes can achieve the HL and its optimal asymptotic coefficient and (ii) when only the standard quantum limit (SQL) is achievable (even with arbitrary adaptive quantum strategies) the optimal asymptotic coefficient of the SQL is also achievable by our codes under slight modifications.

Figure 1

(a) Ancilla-assisted QEC strategy. For example, in Eq. (8), one probe and one noiseless ancilla encode one logical qubit. The HL $[\delta \omega =O(1/N\left)\right]$ is achieved by preparing a logical GHZ state using $N$ logical qubits. (b) Ancilla-free multiprobe QEC strategy. For example, in Eq. (15), $m=\mathrm{\Theta }\left(N\right)$ probes encode one logical qubit. The HL is achieved by preparing a logical GHZ state containing $O\left(1\right)$ number of logical qubits.

Figure 2

(a)–(d) Hamiltonian estimation under Markovian noise. (a) Adaptive quantum strategies (or sequential strategies). (b) Parallel strategies. (c) Ancilla-assisted QEC strategies. (d) Multiprobe QEC strategies that are effectively ancilla free. Here we set the number of probes in a logical qubit $m$ equal to the total number of probes $N$, while in general $N$ can be a multiple of $m$ and the QEC operation will act repeatedly in parallel on each multiprobe logical qubit like in (c). Note that we use slashes over lines to represent ancillary systems whose dimension can be arbitrarily large, and dashed lines to denote small ancillary systems whose dimensions are exponentially small compared to the dimension of the probes. (e), (f) Quantum channel estimation. (e) Parallel strategies. (f) Ancilla-free parallel strategies.

Figure 3

The small-ancilla code is effectively ancilla free because one can set aside $O[log(N\left)\right]$ probes among $N$ total probes to make them function as noiseless ancillas in our construction through an inner layer of QEC, e.g., through the five-qudit code [58, 59]. Note that we say a code achieves the optimal HL if its leading-order term $O\left(N^2\right)$ is optimal, therefore $O[log(N\left)\right]$ probes are negligible in terms of achieving the optimal HL.

Figure 4

Dimension of the ancilla. (a) The dimension of the ancilla in the (1,1)-qutrit code is 3. Here we draw a graph where each vertex represents the probe state $\{|0\rangle ,|1\rangle ,|2\rangle \}$ and there is an edge between two different states $|w\rangle$ and $|w^{\prime }\rangle$ if and only if there exists $S\in \mathcal{S}$ such that $\langle w\left|S\right|w^{\prime }\rangle \ne 0$. We use the color of vertices to represent the (orthonormal) basis of the ancilla. We require adjacent vertices to have different colors, which is a sufficient condition for QEC. Then at least three colors are required. (b) The dimension of the ancilla in the (4,1)-qutrit code is 3. Here we draw a graph where each vertex represents the probe state $\{|0022\rangle ,...,|1111\rangle \}$ (where $...$ includes all permutations of $|0022\rangle$) and there is an edge between two different states $|w\rangle$ and $|w^{\prime }\rangle$ if and only if there exists $S_2\in {\mathcal{S}}_2$ and $\ell \ne {\ell }^{\prime }$ such that $\langle w|S_2^{(\ell ,{\ell }^{\prime })}|w^{\prime }\rangle \ne 0$, where $S_2^{(\ell ,{\ell }^{\prime })}$ means $S_2$ acting on the $\ell \mathrm{th}$ and ${\ell }^{\prime }\mathrm{th}$ probes (or equivalently, if and only if $w$ can be obtained from $w^{\prime }$ by swapping two sites with different states). We require adjacent vertices to have different colors, which is a sufficient condition for QEC. Then at least three colors are required.

Figure 5

Error rate ${\gamma }_{\mathrm{L}}$ as a function of number of probes in the codeword for the small-ancilla code and the ancilla-free random code. We assume the probe evolution is given by the qutrit example discussed in Appendix pp2. Here, the random phase of the random code is picked as the optimal one among 50 samples of random phases that pass a filtering test.

Figure 6

Comparison between the optimal HL and the output QFI of our (effectively) ancilla-free QEC protocols. Here the number of probes in the entire state is equal to the number of probes in each codeword $N=m$. We assume the probe evolution is given by the qutrit example discussed in Appendix pp2. We plot the ideal optimal QFI $F\left[{\rho }_{\omega }\left(t\right)\right]/m^2$ in the noiseless case (in dotted lines) and the output QFIs of the QEC protocol using the small-ancilla code (in solid lines) and the ancilla-free random code (in dashed lines). $m=4,6$ overlaps with the optimal QFI because ${\gamma }_{\mathrm{L}}=0$. The time where the QFI started to decrease has a scaling of $1/{\gamma }_{\mathrm{L}}$. Here, the random phase of the random code is picked as the optimal one among 50 samples of random phases that pass a filtering test, the same as in Fig. 5.