Noise tailoring for robust amplitude estimation

Given a n-qubit Hermitian operator P with eigenvalues ±1 and a quantum state $\vert A\rangle = A \vert 0\rangle^{\otimes n}$, the goal is to estimate the expectation value $\Pi : = \langle A| P |A \rangle$. To do this, the RAE algorithm uses measurement data from enhanced sampling circuits [33], and then performs the classical postprocessing of maximum likelihood estimation (MLE) [38]. An enhanced sampling circuit modifies the trivial SS circuit by adding Grover iterates G, used for quantum amplitude amplification before the measurement of P, see figure 2. For a noisy enhanced sampling circuit with L layers, where each layer resembles G and has a fidelity $f \in [0,1]$, the likelihood of obtaining a measurement bit-string with parity $d \in \{0 ,1\}$ is:

$$\begin{equation} P(d|\Pi,L)=\frac{1}{2}\left(1 + (-1)^d f\,\,{^{L + \frac{1}{2}}}\operatorname{cos}((2L+1)\operatorname{cos}^{-1}\Pi)\right). \end{equation} \tag{ 2 }$$

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The layer fidelity parameter is in-turn related to the nuisance parameter λ of the exponential decay noise model as $f = e^{-\lambda}$, thus incorporating the impact of noise into the above likelihood function.

The classical processing in RAE utilizes a family of enhanced sampling circuits of increasing L, consequently of increasing depth, where a higher depth circuit demonstrate an increase in the inference power [33]. After obtaining a measurement outcome d from an enhanced sampling circuit with L layers, a two-dimensional prior distribution $P(\Pi,\lambda)$ is updated to yield the posterior function [38]:

$$\begin{equation} P(\Pi,\lambda|d)=\frac{P(d|\Pi,\lambda,L)P(\Pi,\lambda)}{\int \operatorname{d}\Pi \operatorname{d}\lambda P(d|\Pi,\lambda,L)P(\Pi,\lambda)}, \end{equation} \tag{ 3 }$$

during the inference process. At the end of the inference process, the estimated expectation is given by the value of Π that maximises the above posterior distribution. Combining likelihood functions for enhanced sampling circuits with different Ls yield an unique estimate with a quantum advantage [43].

It was empirically shown that RAE affords not only a way to improve the precision of one's estimate but also a way to mitigate the bias in one's estimate at least for a two-qubit system [38]. The crucial insight behind this observation is that as long as the noise model describing the quantum hardware is not far from the assumed noise model for RAE, i.e. the exponential decay model, or has the same effect on the algorithm as this assumed noise model, the effect of the noise in MLE is simply to slow the rate of information gain as opposed to biasing the estimates. In this work, we push this insight further and endeavour to tailor the real and complicated noise on a quantum hardware to produce an effective noisy circuit so that f in likelihood function (2) becomes a better approximation to the effect of noise in the estimation process.

For our simulations, we use the model of a noisy superconducting quantum processor and include incoherent errors from amplitude damping and dephasing and coherent errors from residual ZZ interactions between qubits, which are the major sources of error for these processors. Although amplitude and phase damping errors are bad, coherent error is the major performance-limiting factor for quantum algorithms because these errors can accumulate adversarially with increasing circuit depth. Karamlou et al [44] demonstrates that, for superconducting quantum processors, the residual ZZ errors exert a detrimental impact on hybrid algorithms using PQC.

The combined effect of phase and amplitude damping errors is described by a stochastic error channel. For a single-qubit, with decay rate T₁, dephasing rate T₂, and a time-step parameter t_step, this channel is defined using the three Kraus matrices:

$$\begin{align} E_1 &= \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1 - \alpha -\beta} \end{pmatrix}, \nonumber \\ E_2 &= \begin{pmatrix} 0 & \sqrt{\alpha} \\ 0 & 0 \end{pmatrix}, \nonumber \\ E_3 &= \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\beta} \end{pmatrix}, \end{align} \tag{ 4 }$$

where $\alpha = 1 - e^{-\frac{t_\mathrm{step}}{T_1}}$ and $\beta = 1-e^{-\frac{t_\mathrm{step}}{T_2}}$. Extending this formulation to n qubits, the incoherent error channel transforms a n-qubit density matrix ρ into:

$$\begin{equation} \Lambda (\rho) = \sum_{i_1=1}^{i_1=3} \cdots\sum^{i_n=3}_{i_n=1} E_{i_1} \otimes \dots \otimes E_{i_n} \rho E_{i_1}^{\dagger} \otimes \dots \otimes E_{i_n}^{\dagger}. \end{equation} \tag{ 5 }$$

For our simulations, we use the implementation of this error channel provided by IBM's qiskit [45].

The coherent error arising from undesired ZZ interactions between transmon qubits is modelled as a modification to the ideal CNOT gate [44]. These ZZ interactions are produced by small anharmonicities in the qubits. The overall effect of these interactions on a pair of qubits, namely control c and target t, is expressed as:

$$\begin{align} \Xi = \xi_{c, t} (ZZ)_{c,t} + \sum_{i}^{|\mathrm{spec}(c)|} \xi_{c,i} (ZZ)_{c,i} + \sum_{j}^{|\mathrm{spec}(t)|} \xi_{t,j} (ZZ)_{t, j}, \end{align} \tag{ 6 }$$

where $\mathrm{spec}(c)$ and $\mathrm{spec}(t)$ are the sets of qubits coupled to c and t, respectively, with $\xi_{c,i}$ and $\xi_{t,j}$ being the corresponding ZZ interactions. In the presence of these coherent errors, the expression for CNOT over gate time t_gate, in terms of the native cross-resonance gate [46, 47], is modified as:

$$\begin{align} \mathrm{CNOT} &:= e^{-\frac{i \pi}{4}} e^{i\frac{\pi}{4}Z \otimes I} e^{-i\frac{\pi}{4} Z \otimes X } e^{i\frac{\pi}{4}I \otimes X} \nonumber\\ &\longmapsto e^{-\frac{i \pi}{4}} e^{i\frac{\pi}{4}Z \otimes I} e^{i(\Gamma Z \otimes X + \Xi)t_\mathrm{gate} } e^{i\frac{\pi}{4}I \otimes X}, \end{align} \tag{ 7 }$$

where $\Gamma = \frac{\pi}{4t_\mathrm{gate}}$ is the coupling between the control and target qubits.

To understand the effect of incoherent and coherent errors on the RAE algorithm, we investigate how these errors impact the likelihood function (2). To do this, we use the likelihood of getting even parities as:

$$\begin{equation} P(0|\Pi,L) =\frac{1}{2}\left(1 + f\,\,{^{(L + \frac{1}{2})}}\operatorname{cos}((2L+1)\operatorname{cos}^{-1}\Pi)\right), \end{equation} \tag{ 8 }$$

where L = 0 is the SS case. We perform noisy simulations of enhanced sampling circuits with A being the PQC describing a 4-qubit hydrogen molecule (figure 3) and P being the 4-qubit Pauli operator XXXX. In figure 4(a), we present results from simulations incorporating only amplitude and phase damping, with typical T₁ and T₂ values for IBM devices, for a fixed Π and increasing L. We observe that the likelihood values obtained from these simulations yield a good fit to the desired functional form (8), as evident from their perfectly overlapping values and $R^2 = 1$. This implies that despite the noise we should be able to get a good estimate of f and the estimation process could be largely salvaged.

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On the other hand, adding moderate level of coherent errors from ZZ interactions drives the simulation results far away from our assumed functional form with little hope of getting a good estimate of f; see figure 4(b). This observation serves as a strong evidence of how coherent errors impact likelihood values, and can consequently deteriorate the performance of RAE. From these results we infer that with the assumed noise model behind the derivation of the likelihood function (2), RAE will not always offer the sought after sampling power.

Given a n-qubit 'bare' circuit $\mathcal{C}_\mathrm{bare}$, RC produces a set of N random circuits $\{\tilde{\mathcal{C}_\ell}\}~\forall \ell \in \{1, 2, \ldots, N\}$ that are all logically equivalent to $\mathcal{C}_\mathrm{bare}$. Logical equivalence means that all these circuits produce exactly the same wave-function in the noiseless limit. Each $\tilde{\mathcal{C}_\ell}$, which we refer to as a 'random duplicate', is constructed by the following procedure:

• (a)  
  Partition $\mathcal{C}_\mathrm{bare}$ into 'easy' and 'hard' cycles, where we define an easy cycle to be a layer of single-qubit gates and a hard cycle to be a layer of two-qubit entangling gates,
• (b)  
  Uniformly sample from the n-qubit Pauli group $\mathcal{P}_n$ to get an element,
• (c)  
  Insert the element after an easy cycle,
• (d)  
  Insert gates after a hard cycle to ensure the logical equivalence to $\mathcal{C}_\mathrm{bare}$.

Additionally, we combine all adjacent single-qubit gates to preserve the circuit depth. Using a convenient tensor product notation:

$$\begin{equation} \vec{\mathcal{E}_i} =\mathcal{E}_{i1} \otimes \mathcal{E}_{i2} \otimes \cdots \otimes \mathcal{E}_{in}, \end{equation} \tag{ 9 }$$

where $\mathcal{E}_{ij}$ is the single-qubit gate acting on qubit j in an easy cycle i, figure 5 shows the construction and structure of a random duplicate for a 2-qubit circuit.

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In our implementation, the total number of measurements obtained by executing the RC procedure is kept equal to the number of measurements allocated to $\mathcal{C}_\mathrm{bare}$. By defining a tuple $(\mathcal{C}, M)$ for a circuit $\mathcal{C}$ with M measurements, we can thus formulate RC as the map:

$$\begin{equation} f_\mathrm{RC}:(\mathcal{C}_\mathrm{bare},M) \longmapsto (\mathcal{C}_\mathrm{RC},M), \end{equation} \tag{ 10 }$$

where $(\mathcal{C}_\mathrm{RC},M)$ represents a virtual circuit with measurements being the union of measurement outcomes from all random duplicates. For the set $\{(\tilde{\mathcal{C}_\ell}, M_\ell)\}$ and the identity $\sum_{\ell = 1}^{N}M_\ell = M$, we choose:

$$\begin{align} M_\ell &= \lfloor\frac{M}{N}\rfloor,\; \forall \ell \in \{1, 2,\ldots, N-1\},\nonumber\\ M_N &= M-\sum_{\ell=1}^{N-1}M_\ell. \end{align} \tag{ 11 }$$

All these random duplicates in the noisy setting are only distinguished by how the noise affects them since they are all equivalent in the noiseless setting. Then the intuition is that upon running them all on hardware and collecting the results we get bit-strings that behave as though they came from a circuit $\mathcal{C}_\mathrm{RC}$ with no coherent noise but simply stochastic noise.

We study how RC impacts the likelihood of getting even parities (8) from noisy simulations of enhanced sampling circuits (figure 2), with the same A and P as in the previous section. As this likelihood function depends on the exact expectation value Π and number of layers, i.e. Grover iterates, L, our analysis is performed in the following two ways:

• (a)  
  Fix L = 1 and vary $\Pi\in[-1,1]$ by choosing 1000 different values for θ₀ in figure 3,
• (b)  
  Fix $\Pi = 0.2238$ and increase L from 0 to 10.

Furthermore, we generate N = 50 random duplicates for each enhanced sampling circuit.

In figure 6(a), we observe that even in the presence of moderate ZZ interactions with strength $\frac{\xi}{2\pi} = 75$ kHz, RC successfully tailors this coherent error into incoherent error for circuits with one Grover iterate, and consequently brings the likelihood function close to the desired form (8). The tailored circuits fit the function $P(0|\Pi,L = 1)$ closely and yields a layer fidelity of 0.34, signifying that the resultant depolarizing channel is very noisy. As we increase circuit depth beyond one Grover iterate, coherent noise piles up producing no signal, as seen in figure 6(b). The resultant likelihood function is essentially flat and uninformative, with negligible enhanced sampling power; on the other hand for weak coherent noise, i.e. with $\frac{\xi}{2\pi} = 10$ kHz, RC is able to consistently tailor the likelihood values to the desired values for increasing L and the resultant function $P(0|\Pi = 0.2238,L)$ is not very destructive. From these observations we can hypothesize that RC can improve RAE in the presence of coherent noise as long as the noise is not very strong.

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Through simulations of noisy quantum circuits, we study how RC impacts the two features of RAE, namely enhanced sampling for improving RMSE and error mitigation for improving bias of estimates. In the presence of amplitude and phase damping noise and a low magnitude of residual ZZ coupling, RC-assisted RAE achieves slightly lower variances and RMSEs as compared to RAE (figure 8(a)), although both techniques have equivalent scaling of RMSE with runtime (figure 9(a)). In this noise regime, both techniques also have comparable biases because our choice of B cannot resolve changes in biases at the third decimal point.

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By increasing the coherent noise to a moderate value, we observe that RC-assisted RAE performs significantly better than RAE. In this regime, the scaling of RMSE with runtime for RC-assisted RAE exceeds the scaling for RAE up to $L_\mathrm{max} = 7$ (figure 9(b)); this signifies that RC helps RAE to improve RMSE faster. As we have fixed M for increasing L_max, this perceived improvement in RMSE with increasing runtime is solely due to using more layers, and not due to increasing shots. This feature is also evident from figure 8(b) where error bars on the low-biased estimates from our RC-assisted RAE decreases with increasing L_max. Moreover, due to the inherent ability of RC to mitigate the impact of coherent errors on quantum-algorithmic performance, our algorithm is able to significantly reduce bias as compared to RAE in the presence of moderate coherent noise (figure 8(b)).

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Similar to our previous experiments [38], we notice that the estimates oscillate with L_max (figure 8). This might be due to the impact of coherent error being accumulated in some systematic manner as we include additional layers in the estimation process. These oscillations persist in RC-assisted RAE possibly due to the fact that we do not optimize the number of RC duplicates and there might exist some residual coherent errors in the circuits. By allocating the same number of shots for SS and RAE, we observe that RC-assisted RAE also beats SS in the presence of low to moderate coherent error by achieving far less bias and standard deviation (figure 8).

Having established the importance and limitations of RC for the RAE algorithm in simulations, we now test its efficacy on real devices. We showcase results from one publicly-available IBM device: ibmq_belem. For the 4-qubit circuits, RC-assisted RAE has only slight advantage over RAE in terms of reducing bias, but no improvement in precision is observed (figure 10(a)). Consequently, RMSEs from both techniques scale similarly with increasing runtime, where RC-assisted RAE yields slightly less RMSEs than RAE (figure 10(b)), and this reduction of RMSEs is solely due to the corresponding decrease in biases. As these circuits experience huge coherent noise from various sources including residual ZZ couplings and other kinds of crosstalk between qubits, the stochastic noise model achieved after Pauli twirling renders the likelihood function uninformative. In this scenario, 4-qubit enhanced sampling circuits with high L do not contribute significantly in the inference process as they suffer from extreme noise; thus we use $L_\mathrm{max} = 5$ in our plots. Our observation from figure 10(a) suggests a major limitation of RC for RAE: when coherent noise is very high, the effective noise model after RC does not assist RAE and thus we are better off opting for SS.

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For the 2-qubit circuits, although we see a significant reduction in bias with RC-assisted RAE, no noticeable advantage occurs for precision of estimates (figure 11(a)). The impact of coherent error sources on these 2-qubit circuits is lesser than that on 4-qubit circuits; this enables RC to better mitigate noise with even lower number of duplicates. A possible reason for getting lesser biases, both with and without RC, for 4-qubit circuits is that the readout error correction may be less effective for more qubits. These readout errors can introduce additional bias to the estimates, which cannot be tackled by RC. These two-qubit circuits experience a noise regime where RC-assisted RAE beats SS by reducing bias of estimates quite significantly. Consequently, RMSEs obtained from our technique decrease up to $L_\mathrm{max} = 4$, and have lesser values than those achieved by RAE, as evident from figure 11(b).

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In practice, the performance of our RC-assisted RAE algorithm greatly depends on the impact of device noise on the enhanced sampling circuits, and consequently, this algorithm does not always deliver better performance than those from RAE or SS. The effective stochastic noise after RC can be very detrimental when the bare circuit has a high degree of coherent error. This is due to the fact that although RC successfully tailors the likelihood function into the desired form (2), the advantage of RC-assisted RAE ultimately depends on the value of f for this tailored function. A low value for f, corresponding to an effective stochastic noise of high strength, reduces the signal used in the estimation process and thus deteriorates the algorithmic performance.

From our experiments using simulators and real devices, we can thus infer the following:

• (a)  
  RC can help in improving RAE's performance if coherent errors are not high, where the high, moderate and low coherent error regimes can be algorithm dependant.
• (b)  
  Study of optimal number of RC duplicates is needed since this might impact the bias of estimate.