Quantum Computing News and Views

In the past month there have been several interesting developments related to quantum computing:

Spoofing random circuit sampling

Claims of quantum supremacy via random circuit sampling are troublesome because it is so difficult to verify that the quantum processor is working correctly. The Google team used cross entropy benchmarking to verify their original experimental demonstration. The cross entropy benchmarking fidelity is basically a measure of the probability of observing a sequence of bitstrings based on the simulated output probabilities of a circuit. One criticism of this approach is that this fidelity cannot be computed for circuits operating in the classically-intractable quantum supremacy regime. The authors were limited to computing the fidelity for smaller, classically-tractable circuits and comparing its scaling to the expected gate error rates. This approach has several issues explained clearly in a blog post by Gil Kalai discussing a preprint from last year, Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage. 

New attacks of the cross entropy benchmarking fidelity have now appeared as arXiv preprints:

Changhun Oh and collaborators have come up with a method for spoofing the cross entropy benchmarking fidelity in recent Gaussian BosonSampling experiments.

Dorit Aharonov and collaborators show that samples from noisy quantum circuits can be computed classically to a good approximation in polynomial time.

Classical shadows and bosonic modes

Classical shadows continue to be a highly fruitful avenue for a potential near-term quantum advantage. Hsin-Yuan Huang and collaborators have now shown how classical shadows can be used to predict arbitrary quantum processes. To be precise, they give a procedure for estimating observables of a quantum state $\rho$ following application of a completely positive trace-preserving map (describing e.g. evolution of a quantum state in the presence of environmental noise).

When we first become interested in classical shadows almost a year ago a natural question that arose was whether the qubit-based formalism could be translated to bosonic systems. This turns out to be a hard problem because the proof of optimality of shadow methods rests on properties of finite-dimensional vector spaces which do not easily translate to the infinite-dimensional continuous variable quantum systems, discussed in The Curious Nonexistence of Gaussian 2-Designs. 

A team at NIST/University of Maryland now appears to have solved this problem. First, they show that the non-convergent integrals arising in the continouous variable case can be avoided by considering "rigged t-designs" obtained by expanding the Hilbert space to include non-normalizable states. Experimentally-meaningful predictions using this approach can be obtained by regularizing the designs to have a finite maximum or average energy. In a second study the team recasts existing bosonic mode tomography approaches to the classical shadow language to obtain bounds for estimating unknown bosonic states. In practice, however, homodyne tomography performs significantly better than the bounds obtained using classical shadows. These works open up further studies and applications of shadow-based methods to bosonic systems.

Edit: Another paper on continuous variable shadows appeared today: arXiv:2211.07578

IBM's Osprey 433-qubit quantum processor

Last week IBM made several announcements as part of their annual quantum summit, including their new 433-qubit Osprey device. Olivier Ezratty posted a great hype-free analysis of IBM's marketing announcements. In short, large two-qubit gate errors make such large qubit counts useless at this stage, but the advances in integration of the classical control electronics are an important step forward. IBM is still in the lead with the development of superconducting quantum processors. It is also nice to see that they are working towards integrating error mitigation techniques into their software. At present cloud quantum computing providers release devices with atrocious error rates (coughRigetticough), which requires users to spend significant time and money implementing error mitigation techniques in the hope extracting something useful out of their devices.

Bad news for NISQ?

Sitan Chen and collaborators analyze the capabilities of NISQ devices using tools from complexity theory, obtaining a complexity class lying between classical computation and noiseless quantum computation. Their model rules out quantum speedups for certain algorithms including Grover search that are run on near-term noisy quantum devices.

Variational quantum chemistry requires gate-error probabilities below the fault tolerance threshold. The authors compare different variational quantum eigensolvers. Gate errors severely limit the accuracy of energy estimates, making minimizing circuit depths of prime importance. Thus, hardware-efficient ansatzes are preferred over deeper physically-inspired variational circuits. Regardless, supremely low error rates seem to be needed to reach chemical accuracy.

Exponentially tighter bounds on limitations of quantum error mitigation. Quoting from the abstract, "a noisy device must be sampled exponentially many times to estimate expectation values." At first glance, this is good news for cloud quantum processor providers - simply by making your qubits noisier, you can increase your revenue exponentially! However, "there are classical algorithms that exhibit the same scaling in complexity," so classical cloud computing providers will give you much much much better value for money. "If error mitigation is used, ultimately this can only lead to an exponential time quantum algorithm with a better exponent, putting up a strong obstruction to the hope for exponential quantum speedups in this setting."

Towards fault tolerant quantum computing

Daniel Gottesman has a relatively accessible survey on quantum error correcting codes. I was surprised to see that the problem of quantum error correction has not been reduced to an engineering challenge of building bigger devices with the required fidelities; work on understanding the properties and developing better error correction methods more robust against different error types is still an active area of research.

Quantum error correcting codes typically assume that errors are uncorrelated in time and space. In the presence of correlated errors at best you require a lower error rate for quantum error correction to work, and at worst the accumulation of uncorrectable errors will make the result of the computation useless. The Google team have put out a preprint aimed at reducing correlated errors arising from the excitation of higher-order states in their superconducting qubits.

More on the spectral localizer

 At the end of last year Cerjan and Loring put on arXiv a very nice paper studying topological invariants for characterizing metallic systems, based on the properties of a spectral localizer (my summary here). This work was published in Physical Review B a few months ago and was shortly followed by an experimental demonstration of the concept using acoustics in an arXiv preprint, "Revealing Topology in Metals using Experimental Protocols Inspired by K-Theory."

The authors consider an acoustic analogue of the Su-Schrieffer-Heeger model, in which dimerized couplings between nominally-identical resonators create a lattice with a topological band gap. To make the system gapless (metallic), the lattice is coupled to a second gapless array with uniform couplings, forming a two-leg ladder. Because the combined system is gapless, regular topological invariants (based on the existence of an energy gap) cannot be used to predict the existence of robust interface states.

By regularizing the position operator appearing in the spectral localizer (converting the unbounded $X-x_0 I$ to $\tanh ( [X-x_0 I]/\alpha )$, the localizer can be re-interpreted as the Hamiltonian of a related system formed by a suitable modulation of the individual resonators' frequencies; positions $x_0$ where the gap of the spectral localizer vanishes host robust localized modes. The gap closing points are then detected experimentally by measuring the local density of states of the related system described by the localizer Hamiltonian.

One limitation of this approach is that it is based on measuring the properties of a different (but related) system; disorder in the implementation or limitations to the classes of Hamiltonians that can be realized in a particular platform may mean one is unable to faithfully reproduce the localizer of the system of interest. It is therefore interesting to consider how the spectral localizer may be directly measured without construct a separate system, enabling in-situ determination of the topological properties of metallic systems.

One way this might be possible is using nonlinear optical processes such as parametric amplification. To see this, consider the example of one- or two-dimensional lattices with $\textbf{x} = (x,y)$ and the localizer operator

$$ L_{\kappa}(x,y,\omega) = \kappa [(X - x I) \sigma_x + (Y - y I) \sigma_y] + (H - \omega I) \sigma_z, $$

where $\sigma_{x,y,z}$ are Pauli operators, $H$ is the lattice Hamiltonian of interest, and $\kappa$ is a scaling parameter ensuring that the spatial $(x,y)$ and energy $\omega$ scales are comparable. Written in this form, the spectral localizer $L$ resembles a Hamiltonian describing parametric amplification in an inhomogeneously-pumped waveguide lattice [see e.g. Phys. Rev. A 99, 063834 (2019), Nat. Commun. 7, 10779 (2016), or Phys. Rev. X 6, 041026 (2016)].

Specifically, assume the lattice is subject to a two-photon driving term with frequency $\omega_p$ and a site-dependent amplitude described by a diagonal matrix $G$. The two photon drive generates signal and idler photons whose field operators at the $n$th lattice site are $\hat{a}_n$ and $\hat{b}_n$, respectively. Conservation of energy requires their frequencies to satisfy $\omega_a + \omega_b = \omega_p$. For near-degenerate signal and idler photons $\omega_a \approx \omega_b \approx \omega_p/2 \equiv \omega$. The full Hamiltonian describing the parametric amplification is, in a frame rotating at frequency $\omega$,
$$\hat{\mathcal{H}} = \boldsymbol{\hat{a}^{\dagger}} ( H-\omega I ) \boldsymbol{\hat{a}} +  \boldsymbol{\hat{b}^{\dagger}} ( H-\omega I ) \boldsymbol{\hat{b}} + \frac{i}{2} \left(\boldsymbol{\hat{a}^{\dagger}} G \boldsymbol{\hat{b}^{\dagger}} -  \boldsymbol{\hat{a}} G^{\dagger} \boldsymbol{\hat{b}} \right),$$
where $\boldsymbol{\hat{a}} = (\hat{a}_1,...,\hat{a}_N)$ and $\boldsymbol{\hat{b}} = (\hat{b}_1,...,\hat{b}_N)$. The Heisenberg equations of motion $i \partial_t \hat{O} = -[\mathcal{\hat{H}},\hat{O}]$ for the creation and annihilation operators reduce to
$$ i \partial_t \left(\begin{array}{c} \boldsymbol{\hat{a}} \\ \boldsymbol{\hat{b}^{\dagger}} \end{array} \right) = \sigma_z \left( \begin{array}{cc} H - \omega I & iG \\ -iG^{\dagger} & H^*-\omega I \end{array} \right) \left(\begin{array}{c} \boldsymbol{\hat{a}} \\ \boldsymbol{\hat{b}^{\dagger}} \end{array} \right).$$
Assuming a pump profile $G = \gamma [(X - xI) - i (Y - yI)]$, a time reversal-symmetric lattice $H=H^*$, and introducing the field vector $\hat{\psi} = (\hat{a}_1,...\hat{a}_N,\hat{b}_1^{\dagger},...,\hat{b}_N^{\dagger})$, the Heisenberg equations of motion reduce to
$$ i \partial_t \hat{\psi} = L_{i\gamma}(x,y,\omega) \hat{\psi}, $$
corresponding to a spectral localizer with a purely imaginary spatial scale parameter $i\gamma$. The spectrum $\lambda$ of $L_{i\gamma}$ yields the down-converted photon frequencies $\omega_a = \omega + \mathrm{Re}[\lambda]$, $\omega_b = \omega - \mathrm{Re}[\lambda]$, and their gain $\mathrm{Im}[\lambda]$. Thus, properties of the Clifford pseudospectrum may be probed by parametrically driving the lattice of interest, at least for the case of time reversal-symmetric Hamiltonians $H$.

One important difference, however, is that the scale parameter $\kappa$ is purely imaginary under parametric driving. Hence, the spectrum of $L$ comprises purely real eigenvalues (driving weak compared to the detuning) and complex eigenvalues (strong driving). Space and energy are thus treated on an unequal footing, in contrast to the standard Hermitian localizer.

Nevertheless, we can still infer some of the properties of $H$ from the non-Hermitian localizer. Assuming the parametric driving is weak compared to the level spacing of $H$ (which is a reasonable assumption, given that nonlinear optical effects are typically weak), only mode pairs $|\phi_j>$, $|\phi_k>$ of $H$ with resonant energies $\omega_j - \omega = \omega - \omega_k = \nu$ will experience strong amplification. Expanding $L$ in the eigenbasis of $H$ and for compactness defining $Z = X - i Y$ and $z = x-iy$,
$$ L_{i\gamma}(z,\omega) = \left(\begin{array}{cc} \nu & i\gamma (Z_{kj} - z \delta_{kj}) \\ i\gamma (Z_{jk}^* - z^* \delta_{kj}) & \nu \end{array}\right), $$
where $Z_{kj} = <\phi_k| Z | \phi_j>$ and $\delta_{kj}$ is the Kronecker delta function, which appears because different eigenstates of $H$ are orthogonal.

In the case of degenerate down-conversion ($j=k$) the eigenvalues of $L$ reduce to $\lambda = \pm i \gamma |z - <\phi_j|Z|\phi_j>|$. Thus, the gain vanishes when the pump is centred on the resonant mode's centre of mass; degenerate down-conversion will be suppressed, allowing one to infer the mode's frequency and position, and thereby reconstruct the localizer.

More generally, for non-degenerate down-conversion with $j \neq k$, $\lambda = \nu \pm i \gamma <\phi_k|Z|\phi_j>$ and one will detect signal and idler photons with energies $\omega \pm \nu$. The gain is determined by the cross-coupling between the two modes. Recalling $x \sim \partial_k$, this might be sensitive to the Berry connection and/or curvature of near-degenerate modes of $H$, or whatever the real space analogue of these quantities is.

Note that the localizer is most useful for finite disordered systems (since under periodic boundary conditions one can define invariants more easily in terms of the Bloch Hamiltonian). This means it's safe to assume the spectrum of $H$ is non-degenerate.

I'm not sure how well this approach will generalize to other classes of lattices, such as Chern insulators in which time-reversal symmetry must be broken. A similar derivation will hold for $\chi^{(3)}$ nonlinear processes such as spontaneous four wave mixing and third harmonic generation [see e.g. Opt. Exp. 20, 27441 (2012), Nature Photon. 15, 542 (2021), and Phys. Rev. Lett. 123, 103901 (2019)]. Another important problem is to generalize this approach from the simple tight binding approximation to continuous waveguides and photonic crystals, where spatial and spectral correlations of the pump beam affect the efficiency of the down-conversion process.