Fault-tolerant quantum metrology

We introduce the notion of fault-tolerant quantum metrology to overcome noise beyond our control, associated with sensing the parameter, by reducing the noise in operations under our control, associated with preparing and measuring probes and ancillae. To that end, we introduce noise thresholds to quantify the noise resilience of parameter estimation schemes. We demonstrate improved noise thresholds over the non-fault-tolerant schemes. We use quantum Reed-Muller codes to retrieve more information about a single phase parameter being estimated in the presence of full-rank Pauli noise. Using only error detection, as opposed to error correction, allows us to retrieve higher thresholds. We show that better devices, which can be engineered, can enable us to counter larger noise in the field beyond our control. Further improvements in fault-tolerant quantum metrology could be achieved by optimizing in tandem parameter-specific estimation schemes and transversal quantum error correcting codes.

Figure 1

Single-qubit state $|\psi \rangle$ is encoded into multiqubit state ${|\psi \rangle }_L$ in order to detect or correct errors on few qubits. Transversal application of unitary gate $U$ means bitwise application of $U$ on physical qubits of ${|\psi \rangle }_L$. It outputs the encoded state of state $U|\psi \rangle$.

Figure 2

Gate teleportation: All operations outside the dashed box are protected by a code transversal for $\{cX,H,Z\}$. The unitary correction depends on parameter $\varphi$ and since it is not transversal for the code it requires an extra round of distillation.

Figure 3

Three serial quantum metrology protocols for estimating $j$ bits of the phase $\varphi$: (a) Protocol Ia, (b) Protocol Ib, and (c) Protocol Ic. Blue boxes denote the field to be sensed, with its allied noise beyond our control. Orange triangles are inputs and red boxes are measurements, both under our control. The protocols start with the state $|+\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$ probes. Green rhombuses denote fault tolerance interleaved with sensing the field. Filled shapes denote FT implementations.

Figure 4

(a) Scheme with $\gamma =\pi /32$. The allowed region for $\varphi$ is divided into $[0,\pi /2]$ and $[\pi /2,\pi )$. The decision boundaries are the red lines. For $j=1$ the excluded region (gray) is centered around $\pi /2$. For $j=2$, the excluded region (blue) is centered either around $\pi /4$ if $b_1=0$ or around $3\pi /4$ if $b_1=1$. This pattern continues for $j>2$. (b) Noise thresholds for $\gamma =\pi /32$. Red circles, Protocol Ia; blue triangles, Protocol Ib. That the blue line is below the red demonstrates the benefits of FT phase estimation for higher bits. (c) Standard deviation vs resources for $\gamma =\pi /32,p=0.63\%$. Red circles, Protocol Ia; blue triangles, Protocol Ib; markers denote bits starting from 1 and increasing left to right. Improvement from fault tolerance appears in estimating higher bits. See Appendix pp3 for other values of $\gamma .$

Figure 5

Steps 1(i)–1(iv) of Protocol Ib. For $k=1,$ $|\psi \rangle =|+\rangle$ or else the output of previous $k$. $E_{\text{QRM}}$ is the encoding circuit for QRM($1,j+2$), $R_z{\left(\varphi \right)}_L$ is logical (transversal) application of the field, $\left\{{\mathcal{S}}_i^Z\right\}$ are all the $Z$ stabilizer measurements, and $X_L$ is logical $X$ measurement from which we extract the $X$ syndromes.

Figure 6

Relationship between thresholds $p_{\text{th}}$ and $p_{\text{th}}^{\prime }$ for $\gamma =\pi /32$ and $j=4$. Improved threshold for Protocol Ib with device noise (dashed blue) over Ia (solid red/dark gray). Improved threshold of Protocol Ic (solid green/light gray) over Protocol Ib with device noise in subregion enlarged. Protocol Ib without device noise (dotted blue) is provided for reference.

Figure 7

FT application of transversal $R_z\left(\varphi \right)$ using QRM(1,4) and teleportation onto input state $|\psi \rangle$.

Figure 8

Protocol Ic circuit. Operations cnot, $R_z\left(\varphi \right)$, and $X$ measurement are all transversal. Operations $\left\{{\mathcal{S}}_i^Z\right\}$ represent all $Z$ stabilizer measurements. Fault-tolerant measurements and recovery require extra ancillae and correct up to one error.

Figure 9

Parallel phase estimation without fault tolerance.

Figure 10

Estimator of Protocol II. There are three regions: $[0,\pi /2]$, $[\pi /3,2\pi /3]$, and $[\pi /2,\pi ]$. The decision boundaries for $j=1$ are the red lines. There are no excluded regions.

Figure 11

Interrogation noise thresholds. Red circles, Protocol Ia; blue triangles, Protocol Ib without device noise. For the effect of device noise see Fig. 6.

Figure 12

Precision as a function of the number of interrogations, our resource. Red circles, Protocol Ia; blue triangles, Protocol Ib without device noise; markers denote bits. Improvement from fault tolerance is illustrated in estimating higher bits. Noise chosen at the noise thresholds of Protocol Ia which are closer to $0.63\%$: (a) $p=0.639\%$, (b) $p=0.626\%$, and (c) $p=0.619\%$.