Experimental generalized quantum suppression law in Sylvester interferometers

TL;DR

This paper introduces a new suppression law for Sylvester interferometers, experimentally verifies it, and demonstrates its potential for certifying quantum interference in boson sampling, outperforming Fourier interferometers.

Contribution

It proposes and experimentally validates a generalized suppression law for Sylvester interferometers, enhancing quantum interference certification methods.

Findings

Suppression is stronger in Sylvester than Fourier interferometers.

Experimental verification with 4- and 8-mode devices confirms the law.

Potential applications in boson sampling certification.

Abstract

Photonic interference is a key quantum resource for optical quantum computation, and in particular for so-called boson sampling machines. In interferometers with certain symmetries, genuine multiphoton quantum interference effectively suppresses certain sets of events, as in the original Hong-Ou-Mandel effect. Recently, it was shown that some classical and semi-classical models could be ruled out by identifying such suppressions in Fourier interferometers. Here we propose a suppression law suitable for random-input experiments in multimode Sylvester interferometers, and verify it experimentally using 4- and 8-mode integrated interferometers. The observed suppression is stronger than what is observed in Fourier interferometers of the same size, and could be relevant to certification of boson sampling machines and other experiments relying on bosonic interference.

Equations

$$H(2^p)=\left(\begin{array}{cc}H(2^{p-1})& H(2^{p-1})\\ H(2^{p-1})& -H(2^{p-1})\end{array}\right)$$

$${\mathcal{N}}^A(\tilde{r})=\tilde{r}$$

$$\underset{k=1}{\overset{n}{⨁}}\underset{\alpha \in A}{⨁}{S}_{k,\alpha }=1$$

Full text

Introduction. — Scalable, general-purpose quantum computers, once developed, will be able to solve problems thought to be intractable for ordinary computers. Given the significant technological challenges involved, other nearer-term goals for the field were suggested, such as the creation of quantum machines able to beat classical computers in particular computational tasks. One such proposal which has drawn much interest is boson sampling Aaronson and Arkhipov (2011), which relies on multiphoton interference in a random linear interferometer, and whose output statistics are thought to be hard to sample from classically. Due to bosonic interference, the output of those experiments is distributed according to the permanents of complex matrices specifying the interferometer’s design, and the permanent is a function that is notoriously hard to calculate Valiant (1979); Troyansky and Tishby (1996); Aaronson and Arkhipov (2011). This possible path towards a demonstration of quantum computational supremacy has resulted in first experimental implementations of such boson sampling computers Broome et al. (2013); Spring et al. (2013); Tillmann et al. (2013); Crespi et al. (2013); Spagnolo et al. (2013, 2014); Carolan et al. (2014, 2015); Bentivegna et al. (2015); Loredo et al. (2017); Wang et al. (2017).

To investigate the factors behind the computational complexity of these devices, different kinds of input states have been considered. It is known that inputs consisting of distinguishable photons, coherent states, and antisymmetric (fermionic-like) states all result in classically simulable behavior. On the other hand, photon-added coherent states Seshadreesan et al. (2015) and quantum superpositions of coherent states (cat states) Rohde et al. (2015) have been claimed to yield hard-to-simulate outputs. Partially distinguishable photons seem to yield an intermediate regime deserving further investigation Tichy (2015); Tillmann et al. (2015); Shchesnovich (2015). The effect of losses on simulation complexity was also considered Aaronson and Brod (2015); Rahimi-Keshari et al. (2016). Moreover, specific semi-classical states able to reproduce some collective interference effects while being efficiently computable have been identified Tichy et al. (2010).

Other implementations of boson sampling devices serve to improve performance, or were used to discuss how to certify the correct functioning of the device. So-called scattershot boson sampling devices Aar ; Lund et al. (2014); Bentivegna et al. (2015) use simultaneous pumping of several parametric down-conversion sources to result in a random-input version of the original problem. The main advantage is a greatly enhanced generation rate, compared to a single-input implementations using the same sources. While complete certification of large boson sampling devices may turn out to be impossible, a number of proposals for partial certification were made, capable of comparing the experimental outcomes against some physically motivated error models Gogolin et al. (2013); Aaronson and Arkhipov (2014); Spagnolo et al. (2014); Carolan et al. (2014); Bentivegna et al. (2014); Walschaers et al. (2016); Bentivegna et al. (2016); Wang and Duan ; Shchesnovich (2016).

As genuine many-particle interference is required for boson sampling, efforts have been directed at providing stronger evidence of this phenomenon. The simplest such demonstration is the Hong-Ou-Mandel (HOM) effect Hong et al. (1987). Generalisations of this effect have been proposed, to investigate signatures of interference for specific input-output combinations of Fourier Tichy et al. (2010, 2014); Crespi et al. (2016) and Sylvester Crespi (2015) interferometers. These highly-symmetric transformations provide a rich landscape for multi-particle interference to happen, which we investigate here.

In this Letter we discuss and experimentally demonstrate a novel zero-transmission law for Sylvester interferometers. As we will see, indistinguishable photons interfere in these devices so that a certain fraction of all input-output combinations are suppressed. This suggests they may be helpful in identifying multi-photon interference in random-input, scattershot boson sampling experiments. The new zero-transmission law is demonstrated experimentally in Sylvester interferometers with $m=4$ and $m=8$ modes, implemented with the 3-dimensional capabilities of femtosecond laser writing technology.

Suppression law in Sylvester matrices. — Sylvester matrices ($H$) are special constructions of Hadamard matrices of dimension $m=2^{p}$, with $p$ integer, whose elements are only +1 and -1 and can be derived recursively as:

[TABLE]

The associated unitary transformation is a rescaled $U_{\mathrm{S}}^{m}=m^{-1/2}H(m)$. When $p=1$ we have the simplest Sylvester unitary, describing a symmetric 50:50 beam splitter. The celebrated Hong-Ou-Mandel effect Hong et al. (1987) is a first example of suppression law, as a transition between photons in input modes $(1,2)$ and output modes $(1,2)$ is strictly suppressed. In Crespi (2015), more general input states were considered, consisting of $n=2^{q}$ photons, described by a list of mode occupation numbers $(1+nc,\ldots,n+nc)$ with $0\leq c\leq 2^{k}-1$, injected in a $2^{k+q}$-dimensional Sylvester interferometer. It was proven that all outputs whose bitwise sum of binary-represented mode occupation numbers is not zero are strictly suppressed. This suppression law only identifies a small subset of all suppressed input-output combinations.

We now introduce a generalized law that identifies a larger number of suppressed input-output combinations in Sylvester interferometers. A $n$-photon state $|r\rangle$ may be represented by a mode assignment list $\tilde{r}=(\tilde{r}_{1},\tilde{r}_{2},\ldots,\tilde{r}_{n})$, specifying which modes are occupied. It will be useful to represent this state also as a $n\times p$ binary matrix $R$, whose $i$-th row is the binary representation of $\tilde{r}_{i}-1$. Let us call $\mathcal{N}^{A}(R)$ the matrix obtained by negating the columns of $R$ specified by a list of indices $A$. The input-output combinations $|r\rangle\rightarrow|s\rangle$ and $|s\rangle\rightarrow|r\rangle$ are suppressed if some $A$ exists for which the following two conditions are met Not :

[TABLE]

where $\oplus$ denotes the bitwise sum, and $S$ is the binary matrix representation of state $|s\rangle$. In Sup we give a proof of this law (in Sec. I B), as well as a calculation of the asymptotic behavior for the fraction of suppressed events these criteria identify (in Sec. I C). Note that similar criteria were independently reported in Ref. Dittel et al. (2017) while this letter was under completion. For the experimental implementations we report on below, our criteria identify all suppressed events, which are more numerous than the equivalent result for Fourier matrices Tichy et al. (2014) with the same size (See Sec. I D of Sup for more details).

Experimental generalized suppression law. — The generalized suppression law has been tested experimentally in 4-mode and 8-mode Sylvester interferometers, implemented by exploiting a 3D architecture enabled by femtosecond laser writing Crespi et al. (2016) (see Fig. 1). We performed a two-photon scattershot boson sampling experiment Bentivegna et al. (2015) to verify the suppression law, feeding each input port of the 4-mode chip with one heralded photon from a different PDC pair. The output events corresponding to two-photon injection were then post-selected via four-fold coincidence measurements (two heralding detectors and two detectors at the output of the device) for all the $\binom{4}{2}=6$ input-output combinations. The experimental setup adopted for the scattershot approach is shown in Fig. 1. Further details on the generation and detection are reported in Sections II and III of Sup .

The observed two-photon output statistics are shown in Fig. 2, where one can recognize the pattern of peaks and dips predicted by the Sylvester matrix ($U_{\mathrm{S}}^{4}$). An effective figure of merit to estimate if the observed data are compatible with those expected from a fully interfering multiphoton source is the degree of violation $\mathcal{\nu}=N_{\operatorname{forbidden}}/N_{\operatorname{events}}$ Tichy et al. (2014); Crespi et al. (2016), i.e. the ratio between the number of observed events in suppressed input-output combinations and the total number of events. Evaluating the observed violation permits to rule out alternative hypotheses on the nature of the injected state. The simplest alternative hypothesis is that the $n$ photons are distinguishable. A more elaborate alternative model is the Mean Field (MF) state Tichy et al. (2014); Hadzibabic et al. (2004); Cennini et al. (2005), defined as a single-particle state whose wavefunction is macroscopically spread over a set of inputs $A$: $|\psi_{\mathrm{MF}}^{A}\rangle=n^{-1/2}\sum_{r\in A}e^{i\theta_{r}}|j_{r}\rangle,$ where $|j_{r}\rangle$ identifies a single-photon state in mode $j_{r}$, and phases $\theta_{r}$ are randomly chosen from a uniform distribution. Summing the output statistics of $n$ states of this form reproduces some macroscopic interference effects Carolan et al. (2014) of $n$ indistinguishable photons injected in modes $A$. Note that suppression laws are only partially fulfilled by MF states statistics, which confirms the diagnostic power of these tools. In particular, it is easy to check that the expected degree of violation for distinguishable particles and MF states in $U_{S}^{4}$ are respectively $0.66$ and $0.4$ (considering only collision-free two-photon inputs and outputs). The overall experimental violation $\mathcal{\nu}$, obtained by summing all the measured events over all possible input-output combinations, is found to be $\mathcal{\nu}^{(4)}=0.238\pm 0.003$. The measured value, below the two thresholds by respectively $141$ (distinguishable) and $54$ (MF) standard deviations, thus unambiguously excludes both these hypotheses (Fig. 2).

The suppression law has been experimentally verified also using a Sylvester 8-mode chip, where the full set of $\binom{8}{2}=28$ two-photon input states were independently investigated. In Fig. 3 we report the full set of $28\times 28$ experimental probabilities, retrieved from the HOM visibilities measured for all no-collision input-output combinations. In this case, one can show that the theoretical degrees of violation for distinguishable particles and MF states are respectively $0.57$ and $0.31$. The mean value of the violations is $\bar{\nu}^{(8)}=0.115\pm 0.002$, well below the theoretical bounds predicted for MF and distinguishable particles, thus ruling out these alternative hypotheses.

Conclusions and discussion. — We introduced and verified experimentally a novel suppression law pertinent to Sylvester interferometers with inputs of indistinguishable photons. The choice of this unitary transformation ensures a significant fraction of suppressed input-output combinations, even for a larger number of photons and modes. For example, for $n=6$ and $m=16$ our law predicts the suppression of $\sim 10\%$ of the output events, compared to a total fraction of forbidden events of $\sim 37\%$ (computed numerically), against just $\sim 2.3\%$ for the Fourier matrix. This suppression can be used to certify genuine multiphoton interference in scattershot boson sampling experiments, where the full set of possible states is employed. Indeed, random-input generation has been recognized as a promising approach to greatly enhance the event rate in PDC-based scattershot boson sampling experiments. We have shown that the suppression law is able to rule out alternative models which show coarse-grained interference effects, which will be helpful in testing future boson sampling experiments.