Can shallow quantum circuits scramble local noise into global white noise?

In the NISQ-era, we don’t have comprehensive solutions to error correction, which has led the field to develop error mitigation techniques. These techniques aim to extract expected values $\langle O\rangle_{ideal}:=\tr[O\rho_{id}]$ of observables—typically some Hamiltonian of interest—with respect to an ideal noiseless quantum state $\rho_{id}$.

A very simple error model, the global depolarising noise channel, has been very often considered as a relatively good approximation to complex quantum circuits. For qubit states, the channel mixes the ideal, noise-free state with the maximally mixed state $\mathrm{Id}/d$ of dimension $d=2^{N}$ as

Here $\eta\approx F$ is a probability that approximates the fidelity as $F=\eta+(1-\eta)/d$. The white noise channel has been commonly used in the literature for modelling errors in near-term quantum computers [25] and, in particular, it has been shown to be a very good approximation to noise in random circuits [1, 23]. White noise is extremely convenient as it lets the user extract, after rescaling by $\eta$, the ideal expected value of any traceless Hermitian observable $O$ via

Of course, for small fidelities $\eta\ll 1$ the expected value $\tr[O\rho_{wn}]$ requires a significantly increased sampling to suppress shot noise. In this model, the scaling factor $\eta$ is a global property and can be estimated experimentally, e.g., via randomised measurements [25], via extrapolation [26] or via learning-based techniques [27].

Global depolarisation, however, may not be sufficiently accurate model to capture more subtle effects of gate noise and thus rescaling an experimentally estimated expected value yields a biased estimate of the ideal one as $\langle O\rangle_{bias}:=\tr[O\rho]/\eta-\langle O\rangle_{ideal}$. The bias here $\langle O\rangle_{bias}$ is not a global property, i.e., it is specific to each observable, and requires the use of more advanced error mitigation techniques to suppress.

Intuitively, one expects the bias is small for quantum states that are well approximated by a global depolarising model as $\rho\approx\rho_{wn}$ and, indeed, we find a general upper bound in terms of the trace distance as

Here $\lVert O\rVert_{\infty}$ is the operator norm as the absolute largest eigenvalue of the traceless $O$, refer to ref. [28] for a proof. As such, a small trace distance guarantees a small bias and thus indirectly determines the performance of all error mitigation techniques – and further protocols [29, 19].

In this work, we characterise how close noisy quantum states $\rho$ in practical applications approach white noise states $\rho_{wn}$ and consider various types of variational quantum circuits that are typical for NISQ applications. When the above trace distance is small then it guarantees a small bias in expected values which allows us to nearly trivially mitigate the effect of gate noise, i.e., via a simple rescaling.

Another core metric we will consider is the commutator norm between the ideal and noisy quantum states as $\mathcal{E}_{C}:=\lVert[\rho_{id},\rho]\rVert_{1}$, which determines the performance of purification based error mitigation techniques [28] – a small commutator norm has significant practical implications as it guarantees that one can accurately determine expected values using the ESD/VD [21, 22] error mitigation techniques. In particular, independently preparing $n$ copies of the noisy quantum state and applying a derangement circuit to entangle the copies, allows one to estimate the expected value

The approach is very NISQ-friendly [30, 31] and its approximation error $\mathcal{E}_{ESD}$ approaches in exponential order a noise floor as we increase the number of copies $n$ [21]; This noise floor is determined generally by the commutator norm $\mathcal{E}_{C}$ but in the most typical applications of preparing eigenstates, the noise floor is quadratically smaller as $\mathcal{E}_{C}^{2}$ [28].

Note that this commutator can vanish even if the quantum state is very far from a white noise state, in fact it generally vanishes when $\rho_{id}$ approximates an eigenvector of $\rho$. When a state is close to the white noise approximation then a small commutator norm is guaranteed, however, we demonstrate that the latter is a much less stringent condition and a much better approximation in practice than the former: in all instances we find that the commutator norm is significantly smaller than the trace distance from white noise states.

Recall that any quantum state of dimension $d$ can be represented via its density matrix $\rho$ that generally admits the spectral decomposition as

where we focus on $N$-qubit systems of dimension $d=2^{N}$. Here $\lambda_{k}$ are non-negative eigenvalues and $\ket{\psi_{k}}$ are eigenvectors. Since $\sum_{i}\lambda_{i}=1$, the spectrum $\underline{\lambda}$ is also interpreted as a probability distribution.

If $\rho$ is prepared by a perfect, noise-free unitary circuit, only one eigenvalue is different from zero and the corresponding eigenvector is the ideal quantum state as $\ket{\psi_{id}}$. In contrast, an imperfect circuit prepares a $\rho$ that has more than one non-zero eigenvalues and is thus a probabilistic mixture of the pure quantum states $\ket{\psi_{k}}$, e.g., due to interactions with a surrounding environment. In fact, noisy quantum circuits that we typically encounter in practice produce quite particular structure of the eigenvalue distribution: one dominant component that approximates the ideal quantum state $\ket{\psi_{1}}\approx\ket{\psi_{id}}$ mixed with an exponentially growing number of “error” eigenvectors that have small eigenvalues. White noise is the limiting case where non-dominant eigenvalues are exponentially small $\propto 1/d$ and $\ket{\psi_{1}}\approx\ket{\psi_{id}}$.

The quality of the noisy quantum state is then defined by the probability of the ideal quantum state as the fidelity $F:=\expectationvalue{\rho}{\psi_{id}}$; We show in Appendix A that for any quantum state it approaches the dominant eigenvalue $\lambda_{1}$ as

where we compute the error term analytically in terms of the commutator norm $\mathcal{E}_{C}=\lVert[\rho_{id},\rho]\rVert_{1}$ from Section I.1. This property is actually completely general and applies to any density matrix.

Most typical noise models used in practice, such as local depolarising or dephasing noise, admit the following probabilistic interpretation: a noisy gate operation $\Phi(\rho)$ can be interpreted as a mixture of the noise-free operation $U$ that happens with probability $1-\epsilon$ and an error contribution as

Here $U_{k}$ is the $k^{th}$ ideal quantum gate and the completely positive trace-preserving (CPTP) map $\Phi_{err}$ happens with probability $\epsilon$ and accounts for all error events during the execution of a gate. A quantum circuit is then a composition of a series of $\nu$ such quantum gates which prepares the convex combination as

Here $\rho_{id}:=\outerproduct{\psi_{id}}{\psi_{id}}$ is the ideal noise-free state, $\rho_{err}$ is an error density matrix and $\eta=(1-\epsilon)^{\nu}$ is the probability that none of the gates have undergone errors. This probability actually [28, 23] approximates the fidelity $F:=\expectationvalue{\rho}{\psi_{id}}$ given the noise model in Eq. 7 as

Here we approximate $(1-\xi/\nu)^{\nu}=e^{-\xi}+O(\epsilon^{2}/\nu)$ for small $\epsilon$ and large $\nu$ where $\xi:=\epsilon\nu$ is the circuit error rate as the expected number of errors in a circuit. In practice the approximation error $\mathcal{E}_{F}=\expectationvalue{\rho_{err}}{\psi_{id}}$ is typically small and in the limiting case of white noise it decreases exponentially as $\mathcal{E}_{F}=1/d$ due to $\rho_{err}=\mathrm{Id}/d$.

Assuming sufficiently deep, complex circuits, ref. [28] obtained an approximate bound for the commutator between the ideal and noisy quantum states as

This bound confirms that as we increase the number of quantum gates $\nu$ in a circuit but keeping the circuit error rate $\xi$ constant, the commutator norm decreases as $\propto 1/\sqrt{\nu}$ [28]. Furthermore, this function closely resembles to Eq. 1 which is a central aim of this work to explore.

Random circuits have enabled quantum supremacy experiments using noisy quantum computers for two primary reasons: (a) the outputs of these circuits are hard to simulate classically and (b) they render local noise into global white noise [1], hence introducing only a trivial bias to the ideal probability distribution similarly as in Section I.1.

Ref [23] considered random circuits consisting of $s$ two-qubit gates, each of which undergoes two single qubit (depolarising) errors each with probability $\tilde{\epsilon}$ (assuming single-qubit gates are noiseless). We can relate this to our model by identifying the local noise after each two-qubit gate with the error event in Eq. 7 via the probability $\epsilon=1-(1-\tilde{\epsilon})^{2}=2\tilde{\epsilon}-\tilde{\epsilon}^{2}$. Ref [23] then established the fidelity $\tilde{F}$ of the quantum state which one obtains from a noisy cross-entropy score as

This coincides with our approximation from Eq. 9 up to an additive error in the exponent which, however, diminishes for low gate error rates. In the following we will thus assume $F\equiv\tilde{F}$.

Measuring these noisy states in a the standard measurement basis $\{|j\rangle\}_{j=1}^{d}$ produces a noisy probability distribution $\tilde{p}_{noisy}(j)=\langle j|\rho|j\rangle$. Ref. [23] established that this probability distribution rapidly approaches the white noise approximation $\tilde{p}_{wn}=Fp_{id}+(1-F)p_{unif}$. In particular, the total variation distance (via the $l_{1}$ norm $\lVert x\rVert_{1}=\sum_{i}|x_{i}|$) between the two probability distributions is upper bounded as

This expression is formally identical to the bound on the commutator norm in Eq. 10; Indeed if the noise in the quantum state approaches a white noise approximation, it implies that the commutator norm must also vanish in the same order.

On the other hand, the reverse is not necessarily true as Eq. 11 is a stronger condition than Eq. 10 as the latter only guarantees that the dominant eigenvector approaches $\ket{\psi_{1}}\approx\ket{\psi_{id}}$ but does not imply anything about the eigenvalue distribution of $\rho$ or $\rho_{err}$.

In the NISQ-era comprehensive error correction will not be feasible and thus hope is primarily based on variational quantum algorithms [11, 12, 13, 33, 34]. In this paradigm a shallow, parametrised quantum circuit is used to prepare a parametrised quantum state that aims to approximate the solution to a given problem, typically the ground state of a problem Hamiltonian. Due to its shallow depth the ansatz circuit is believed to be error robust and its tractable parametrisation allows to explore the Hilbert space near the solution. On the other hand, such circuits are structurally quite different than random quantum circuits and it was already raised in ref. [23] whether error bounds on the white noise approximation extend to these shallow quantum circuits.

We simulate such quantum circuits under the effect of local depolarising noise – while note that a broad class of local coherent and incoherent error models can effectively be transformed into local depolarising noise using, e.g., twirling techniques or randomised compiling [35, 36, 37, 38]. We analyse the resulting noisy density matrix $\rho$ by calculating the following two quantities. First, we quantify ‘closeness’ to a white noise state from Eq. 2 by computing uniformity measure $W$ as the $l_{1}$-distance between the uniform distribution and the non-dominant eigenvalues of the output state as

which only depends on spectral properties of the quantum state and can thus be computed straightforwardly. We show in 2 that $W$ is proportional to the trace distance from a white noise quantum state as

uo to a bounded error $\mathcal{E}_{w}$. The uniformity measure $W$ thus determines the bias in estimating any traceless expected value as discussed in Section I.1.

Second, we calculate the commutator norm $\mathcal{E}_{C}$ from Section I.1 relative to $1-\lambda_{1}$ as

which we relate to the commutator norm between the “error part” of the state $\rho_{err}$ and the ideal quantum state $\rho_{id}$ in Lemma 1. In the following, we will refer to $C$ as the commutator norm – and recall that it determines the ultimate performance of purification-based error mitigation as discussed in Section I.1.

We first consider a Strong Entangling ansatz (SEA): it is built of alternating layers of parametrised single-qubit rotations followed by a series of nearest-neighbour CNOT gates as illustrated in Fig. 5 – and assume a local depolarising noise with probability $\epsilon$. We simulate random quantum circuits by randomly generating parameters $|\theta_{k}|\leq 2\pi$ – note that these circuits are not necessarily Haar-random distributed and thus results in Section II.3 do not necessarily apply.

We simulate $10$-qubit circuits and in Fig. 1 (a) we plot the eigenvalue uniformity $W(\nu)$ while in Fig. 1 (b) we plot the commutator norm $C(\nu)$ for an increasing number $\nu$ of quantum gates – all datapoints are averages over ten random seeds. These results surprisingly well recover the expected behaviour of random quantum circuits as for small error rates $\epsilon\rightarrow 0$ both quantities $W(\nu)$ and $C(\nu)$ can be approximated by the function from Eq. 1 as we now discuss.

In Section II.3 we stated bounds of ref. [23] on the distance between $\tilde{p}_{noisy}$ and $\tilde{p}_{wn}$. Based on the assumption that these bounds also apply to the probability distributions $p_{noisy}=\bra{\psi_{k}}\rho\ket{\psi_{k}}$ and $p_{wn}:=\bra{\psi_{k}}\rho_{wn}\ket{\psi_{k}}$ we derive in 4 the approximate bound on the eigenvalue uniformity as

Furthermore, by combining Eq. 14 and the bound in Eq. 10 we find that the commutator norm $C$ is similarly bounded by the same function. On the other, Fig. 1 (b) suggests that the commutator norm decreases with a larger polynomial degree and thus we approximate both $W(\nu)$ and $C(\nu)$ using the function

where we fit the two parameters $\alpha$ and $\beta$ to our simulated dataset. The second equation above is an expansion for small circuit error rates $\xi$ as detailed in Section A.2.1. Indeed, in Fig. 1 (blue circles) for small $\epsilon\rightarrow 0$ we observe a nearly linear function in the log-log plot in Fig. 1 and thus remarkably well recover the theoretical bounds with the polynomial power approaching $b\rightarrow 1/2$.

Furthermore, comparing Fig. 1 (b, blue circles) and Fig. 1 (a, blue circles) suggest that the commutator norm has both a significantly smaller absolute value (smaller $\alpha$) and decreases at a faster polynomial rate (larger beta) than the uniformity measure. In fact, the commutator norm is more than two orders of magnitude smaller than the uniformity measure which suggests that even when $\rho_{err}$ is not approximated well by a white noise state it, nevertheless, almost commutes with the ideal pure state $\rho_{id}$.

We finally consider how the absolute factor $\alpha$ depends on the number of qubits: we perform simulations at a small error rate $\epsilon\rightarrow 0$ and fit our model function $\alpha\nu^{\beta}$ to extract $\alpha(N)$ for an increasing number of qubits. The results are plotted in Fig. 7 (e) and suggest that the prefactor $\alpha(N)$ initially grows slowly but then saturates while note that a polylogarithmic depth is sufficient to reach anticoncentration [23].

Theoretical results guarantee that the SEL ansatz initialised at random parameters approach for an increasing depth unitary 2-designs thereby reproducing properties of random quantum circuits [39, 40]. It is thus not surprising that the model introduced in Section II.3 gives a remarkably good agreement between the SEL ansatz (dots on in Fig. 1) and genuine random circuits (fits as continuous lines in Fig. 1).

Here we consider the Hamiltonian Variational Ansatz (HVA) [41, 42] at more practical parameter settings: The HVA has the advantage that we can efficiently obtain parameters that increasingly better (as we increase the ansatz depth) approximate the ground state of a problem Hamiltonian – we will refer to these as VQE parameters. We also want to compare this circuit against random circuits and thus also simulate the HVA such that every gate receives a random parameter as detailed in Section B.1.

While the VQE parameter settings capture the relevant behaviour in practice as one approaches a solution, the random parameters are more relevant, e.g., at the early stages of a VQE parameter optimisation. Furthermore, as the circuit is entirely composed by Pauli terms in the problem Hamiltonian, the dimensionality of its dynamical Lie algebra is entirely determined by the problem Hamiltonian in contrast to an exponentially growing algebra of the SEL ansatz [24].

We first consider a VQE problem of finding the ground state of the 1-dimensional XXX spin-chain model. We construct the HVA ansatz from Section III.3 for this problem Hamiltonian as a sum $\mathcal{H}_{\text{XXX}}=\mathcal{H}_{0}+\mathcal{H}_{1}$ as

The Pauli operators $XX$, $YY$ and $ZZ$ determine couplings between nearest neighbour spins in a 1-D chain and we choose them to be of unit strength. Furthermore, $Z_{k}$ are local on-site interactions $|\Delta_{k}|\leq 1$ that were generate uniformly randomly such that the Hamiltonian has a non-trivial ground state.

First, we simulate the HVA ansatz for $N=10$ qubits with randomly generated circuit parameters as $|\theta_{k}|\leq 2\pi$ and plot results for an increasing number of quantum gates in Fig. 2 (a, c). We a find similar behaviour for the eigenvalue uniformity $W(\nu)$ as with random SEL circuits in Fig. 1 (a) and obtain a reasonably good fit for $\epsilon\rightarrow 0$ using our model function from Eq. 15. The commutator norm in Fig. 2 (c) is again significantly smaller in magnitude than the uniformity measure and decreases faster with a higher polynomial order similarly to as with the random SEL ansatz in Fig. 1 (b) .

Second, in Fig. 2 (b,d) we simulate the ansatz at the VQE parameters that approximate the ground state. Since the ansatz parameters become very small as one approaches an adiabatic evolution, it is not surprising that the output density matrix is not well-approximated by a white noise state: the uniformity measure is very large in Fig. 2 (b). The commutator norm in Fig. 2 (d) again, is significantly smaller than $W(\nu)$ and although it appears to slowly grow with $\nu$, it appears to decrease for $\nu\rightarrow\infty$. This agrees with observations of ref. [28] that the circuits need not be random for the commutator to be sufficiently small in practice.

Furthermore, in Fig. 7 (a, b) we investigate the dependence on $N$ and find that the prefactor $\alpha$ grows slowly and appears to saturate for $N\geq 10$.

In the next example we consider the transverse-field Ising (TFI) model $\mathcal{H}_{\text{TFI}}=\mathcal{H}_{0}+\mathcal{H}_{1}$ using constant on-site interactions $h_{i}=1$ and randomly generated coupling strengths $|J_{i}|\leq 1$ as

We first simulate the HVA ansatz with random variational parameters in Fig. 3 (a, c). While at small error rates $\epsilon\rightarrow 0$ Fig. 3 (a, blue) can be fitted well with our polynomial approximation form Eq. 15, we observe that the eigenvalue uniformity $W(\nu)$ in Fig. 3 (a, blue) decreases with a small polynomial degree.

Indeed, as the HVA ansatz is specific to a particular Hamiltonian, its dynamical Lie algebra may have a low dimensionality [24] resulting in a limited ability to scramble local noise into white noise; this explains why in Fig. 3 (a) the uniformity measure decreases more slowly, i.e., in a smaller polynomial order, than random circuits. For this reason, we additionally simulate in Fig. 6 the TFI-HVA ansatz but with adding $R_{z}$ gates in each layer whose generator is not contained in the problem Hamiltonian. The increased dimensionality of the dynamic Lie algebra, indeed, improves scrambling as the white noise approximation is clearly better in Fig. 6 – while note that the increased dimensionality may also lead to exponential inefficiencies in training the circuit [24].

In stark contrast to the case of the uniformity measure $W(\nu)$, we find that the commutator norm in Fig. 3 (c, blue) decreases substantially for an increasing $\nu$ despite the low dimensionality of the Lie algebra. This nicely demonstrates that a small commutator norm is a much more relaxed condition than white noise as the latter requires that the noise is fully scrambled in the entirety of the exponentially large Hilbert space. Finally, we simulate the TFI circuits at VQE parameters and find qualitatively the same behaviour as in the case of the XXX problem.

We consider a 6-qubit Lithium Hydride (LiH) Hamiltonian in the Jordan-Wigner encoding as a linear combination of non-local Pauli strings $P_{k}\in\{\mathrm{Id},X,Y,Z\}^{\otimes N}$ as

We construct the HVA ansatz by splitting this Hamiltonian into two parts with $\mathcal{H}_{0}$ being composed of the diagonal Pauli terms in Eq. 17 while $\mathcal{H}_{1}$ composed of non-diagonal Pauli strings.

Such chemical Hamiltonians typically have a very large number of terms with $r_{h}\gg 1$ but a significant fraction only have small weights $h_{k}$ thus the HVA would have a large number of gates with only very small rotation angles. For these reasons we construct a more efficient circuit whose basic building blocks are constructed using sparse compilation techniques [43, 44]: Each single layer in the HVA ansatz consists of gates that correspond to $100$ randomly selected terms of the Hamiltonian with sampling probabilities $p_{k}\propto|h_{k}|$ proportional to the Pauli coefficients. This approach has the added benefit that it makes the circuit structure random as opposed to the fixed structures in Section III.4 and in Section III.5.

Results shown in Fig. 4 (a,c) agree with our findings from the previous sections: at randomly chosen circuit parameters the uniformity measure decreases according to Eq. 15; the commutator norm similarly decreases but in a higher polynomial order while its absolute value is smaller by at least an order of magnitude. In contrast, Fig. 4 (b) suggests that the errors are not well approximated by white noise with a large and non-decreasing $W(\nu)\approx 0.5$. Furthermore, Fig. 4 (b) again confirms that despite white noise is not a good approximation, the commutator norm is small in absolute value, i.e., $\approx 10^{-3}$ in the practically relevant region. This guarantees a very good performance of the ESD/VD error mitigation techniques sufficient for nearly all practical purposes.