Linear optical quantum computing was one of the earliest success stories in quantum computing research (O’Brien, Reference O’Brien2007). Since the groundbreaking paper of Knill, Laflamme and Milburn in 2000, which demonstrated that linear optics and post-selected measurements (Knill et al., Reference Knill, Laflamme and Milburn2001) could be used to realise universal scalable quantum logic in discrete photonics, experimental demonstrations of small-scale quantum algorithms have become routine around the world (Bouwmeester et al., Reference Bouwmeester, Pan, Daniell, Weinfurter and Zeilinger1999; Lu et al., Reference Lu, Zhou, Gühne, Gao, Zhang, Yuan, Goebel, Yang and Pan2007; Lanyon et al., Reference Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist and White2007; Alberto et al., Reference Peruzzo, Lobino, Matthews, Matsuda, Politi, Poulios, Zhou, Lahini, Ismail, Wörhoff, Bromberg, Silberberg, Thompson and OBrien2010; Stefanie et al., Reference Stefanie, Elham, Anne, Joseph, Anton and Philip2012; Jacques et al., Reference Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien and Laing2015). Bulk optical experiments in the laboratory were replaced with more stable integrated optical photonic chips and algorithms consisting of up to 12 photonic qubits have been performed in a controlled manner (Zhong et al., Reference Zhong, Li, Li, Peng, Su, Hu, He, Ding, Zhang, Li, Zhang, Wang, You, Wang, Jiang, Li, Chen, Liu, Lu and Pan2018). In recent years, photonic quantum computing systems have been adopted by several startups worldwide (Kaczmarek et al., Reference Kaczmarek, Nunn, Poem and Peizpour2020; Bartolucci et al., Reference Bartolucci, Bichall, Bombin, Cabal, Dawson, Gimeno-Segovia, Johnston, Kieling, Nickerson, Pant, Pastawski, Rudolph and Sparrow2021), with the promise of building scalable single-photon-based quantum computers (Rudolph, Reference Rudolph2017).
However, quantum computing platforms based on single photons have traditionally suffered from the reliable generation of on-demand single photons. The vast majority of experimental demonstrations of optical quantum computing use sources that are probabilistic, producing individual photons from the down-conversion of low amplitude coherent states from pulsed lasers (henceforth called weak laser pulse) (Jacques et al., Reference Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien and Laing2015). These non-deterministic sources are a major limiting factor for the demonstration of larger quantum algorithms in photonic platforms when millions or more photons are required in a temporally synchronous manner. The push to build a scalable quantum computing system in optics is dependent on solving the source problem and constructing devices that can produce high-quality, identical, on-demand single photons.
In this paper, we detail a design for a new type of multiplexed source that exploits a specific property of integrated optical switches, namely their asymmetric switching properties. Single-photon switches, based on Mach-Zehnder interferometers and phase modulators, often have anti-symmetric phase profiles as a function of the control parameter – that is they effectively have different time scales associated with “turning-on” and “turning-off” (Liu et al., Reference Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo2022). For example, using thermal pads to phase modulate a wave-guide in doped silicon can be switched “on” quickly by adding thermal energy to the system, but removing that thermal energy to switch “off” the modulator takes up to five times as long (Liu et al., Reference Liu, Tian, Lu and Feng2021; Sabouri et al., Reference Sabouri, Angel Mendoza Velasco, Catuneanu, Namdari and Jamshidi2021).
Our new design – dubbed the racetrack source – considers this effect and allows for the construction of a multiplexed source where individual switches are only used once. This allows for fast multiplexing using hardware switching that may be slow. We perform detailed error modelling and illustrate that a high-probability multiplexed source can be constructed.
Background
Multiple approaches have been proposed for building a device with the capability of producing high-quality photons on-demand. They can be broken down into two broad categories.
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Single-photon emitters: Matter-based quantum systems (such as atoms or quantum dots) that can be controlled and pulsed to emit single photons deterministically as the result of an energy transition (Ding et al., Reference Ding, He, Duan, Gregersen, Chen, Unsleber, Maier, Schneider, Kamp, Höfling, Lu and Pan2016; Aharonovich et al., Reference Aharonovich, Englund and Toth2016; Senellart et al., Reference Senellart, Solomon and White2017; Huber et al., Reference Huber, Reindl, Huo, Huang, Wildmann, Schmidt, Rastelli and Trotta2017; Awschalom et al., Reference Awschalom, Hanson, Wrachtrup and Zhou2018; Ding et al., Reference Ding, Guo, Xu, Liu, G-Yn, Zhao and Ge2023).
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Multiplexing: Combining a large number of probabilistic sources with an active switching network to route successful photon generation events to the output of the device (Pittman et al., Reference Pittman, Jacobs and Franson2002; Ma et al., Reference Ma, Zotter, Kofler, Jennewein and Zeilinger2011; Mower and Englund, Reference Mower and Englund2011; Collins et al., Reference Collins, Xiong, Rey, Vo, He, Shahnia, Reardon, Krauss, Steel, Clark and Eggleton2013; Bonneau et al., Reference Bonneau, Mendoza, OBrien and Thompson2015; Kaneda and Kwiat, Reference Kaneda and Kwiat2019; Meyer-Scott et al., Reference Meyer-Scott, Silberhorn and Migdall2020).
While single-photon emitters may ultimately prove to be a more effective means of producing photons for quantum computing and communication applications in the future, they currently come with several drawbacks. The first is the ability to produce identical photons in artificial atoms such as quantum dots. For individual photons to be useful in optical quantum processing chipsets, they must interfere with high fidelity when mixed together on linear optical elements, such as beamsplitters. For this to occur, individual photons must be close to 100% indistinguishable in all degrees of freedom. This is still difficult to achieve. The physical fabrication of the solid-state emitter, its local environment and control inaccuracies all influence the ability to produce individual photons from distinct emitter sources that interfere with high accuracy on linear optical components. Additionally, the integration of these components into linear optical quantum technology often diminishes the benefits of the optical platform as they are comparatively difficult and costly to fabricate at scale, can require complex infrastructure, such as dilution refrigeration cooling to operate, and are not always compatible with telecom frequencies when utilising long-range optical fibre systems as low-loss quantum memories for scalable designs (Bombin et al., Reference Bombin, Kim, Litinski, Nickerson, Pant, Pastawski, Roberts and Rudolph2021).
The current method of choice for photon production remains probabilistic down-conversion processes such as in Spontaneous Parametric Down Conversion (SPDC) or Spontaneous Four Wave Mixing (SFWM) sources (Jacques et al., Reference Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien and Laing2015) the latter of which has already been incorporated into integrated optical chipsets and experimentally utilised). These sources take a weak coherent laser source of frequency (ω) and, through a weak optical nonlinearity, probabilistically generate a pair of lower frequency photons (ω 1 + ω 2 = ω). The probability of success for these single-photon sources is dictated by the power used in the pump laser. The general output Gaussian state given a weak laser pulse input can be described by the sum over Fock states,
$$\sqrt {1 - |\zeta {|^2}} \left( {|vac{\rangle _s}|vac{\rangle _i} + \sum\limits_{n = 1}^\infty {{\zeta ^N}|n{\rangle _s}|n{\rangle _i}} } \right).$$
The subscripts s and i refer to the signal and idler photons and |ζ| < 1 parameterises the strength of the non-linearity in the source and the power used in the pump, |ζ| = tanh (cP), where P is the power in mW and c is the coupling constant of the optical non-linearity in units of mW−1/2 (Walls and Milburn, Reference Walls and Milburn2012).
The output of the source is consequently not a perfect single photon; we instead always get a superposition of the vacuum state and all Fock states. As |ζ| < 1, the higher-order Fock terms drop off exponentially. Therefore, good single-photon sources are highly probabilistic. This creates a scaling problem. For the production of two photons at the same time, two down-conversion sources must succeed at the same time, occurring with a probability of p 2. If each source succeeds with a probability of 1/100, then two photons can only be produced with a probability of 1/10,000. This continues to decrease exponentially, such that producing N photons at the same time for a quantum algorithm occurs with a probability of p N .
The solution to this problem is the concept of multiplexing: A large number of probabilistic sources are integrated with an active switching network so that when the device is triggered, on average, one of the probabilistic sources will be successful and the switching network can be configured, in real-time, such that the resultant photon can be routed to the output of the device. However, the utility of this device to produce on-demand single photons is contingent on very accurate and fast single-photon switches. These switches are difficult to fabricate reliably, and often a choice needs to be made between fast, lossy, switches versus precise slow switches. This is often due to the intrinsic asymmetry of the switches themselves. While switches built, for example, from waveguide thermal pads (Jacques et al., Reference Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien and Laing2015) can be switched on very quickly (by dumping a large amount of thermal energy onto a chip in a very short time), they cannot be switched off quickly – essentially the system has to be allowed to cool through thermal dissipation. This limits either the speed or reliability of the switch itself and limits its use in a multiplexed source.
Research has been ongoing to use different physical effects to induce the modulation needed to build a single-photon switch – most notably electro-optical modulation in materials such as Lithium Niobate (Wooten et al., Reference Wooten, Kissa, Yi-Yan, Murphy, Lafaw, Hallemeier and Maack2000) – but these systems are still in their technological infancy and are not straightforwardly compatible with the CMOS manufacturing. This is important as CMOS compatibility and the ability to leverage global silicon manufacturing infrastructure for quantum technology is a big advantage for the utility of optics as a quantum hardware system.
In addition to intrinsic waveguide loss, inaccurate switching in these multiplexed sources is a large contributing factor to photon loss. High loss rates or slow production of on-demand photons renders multiplexed photon sources effectively useless in the current generation of integrated optical systems, especially without the existence of high-quality and low-loss single-photon memories. Photons need to be produced quickly, such that small amounts of fibre loops are sufficient to temporally synchronise photons as input to integrated optical computing chips and the pseudo-determinism of such a device has to be high to move into a regime where the probability of generating N photons for an experiment scales at worst polynomially rather than exponentially in N.
Our goal is to redesign a multiplexed switch specifically with this asymmetry in mind, using the CMOS-compatible thermal pad system to prototype a new design that operates each switch in the multiplexed system in a “one-way” fashion. As each switch is fast and accurate when switched “on” from a previously “off” position, we will construct a multiplexed switch where this is the only operation allowed for each switch in the device. This leads to a source design that resembles a racetrack, with an array of switches coupling an inner waveguide loop to an outer loop that contains an array of probabilistic sources and heralding detectors. The key insight is that each switch in the system can only be used ONCE, sidestepping the fundamental problem that makes switches in a multiplexed source either too unreliable or too slow.
In this work, we present a novel multiplexed single-photon source that addresses asymmetric configuration delays and corresponding error characteristics to produce high-quality photons with a high probability.
Asymmetric switch
In our proposed single-photon source each switch is configured at most twice. This enables the exploitation of asymmetric configuration delays for switches with a specific switching profile as sketched in Figure 1.
Figure 1. Schematic switching profile of switches based on thermo-optic effect.
For currently available switches based on thermo-optic effects, the switch configuration can often be changed once quickly for example, by turning on a heat pad or by active cooling (Liu et al., Reference Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo2022). However, as soon as the switch is configured once, reverting to the previous configuration requires a larger configuration delay. According to the literature, the configuration delay T sr incurred by turning on a heat pad in thermo-optic switches may be approximately 3 times smaller to 4 times larger than T sf , the configuration delay incurred by cooling down the heat pad (Liu et al., Reference Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo2022). Regardless of T sr or T sf requiring more time, the polarity of the switches can be reversed to exploit the asymmetric configuration delays for switching in the proposed single-photon source. For the remainder of the paper, we assume that the time required to turn on a heat pad in a thermo-optic switch requires significantly less time than its cooldown.
Thermo-optic switches
A single-photon switch is a two-input, two-output device that allows for a single photon to be routed from either of the two inputs to either of the two outputs. It has a generic structure shown in Figure 2, which is drawn in the context of an integrated optics chipset.
Figure 2. Switch based on thermo-optic effects with two thermal pads that can be activated individually.
In Figure 2, we have three components, two directional couplers that work as single-photon beamsplitters and a single device, known as a phase shifter, that acts to impart a relative phase on any photon that passes through the waveguides underneath. See for example, Walls and Milburn (Reference Walls and Milburn2008) for a derivation of the transmission function of this Mach-Zehnder modulator.
Thermo-optic switches can be used in Figure 2 that contain two thermal pads, each applicable to one of the connected waveguides separately. The induced phase shift θ can be manipulated by each heat pad individually. The initial switch configuration can be reverted in time T sr by activating the second heat pad. However, changing the switch configuration a third time would then require the cooldown of one or both heat pads, leading to the delay T sf . The manufacturing of the proposed double-padded switches has been demonstrated in Liu et al. (Reference Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo2022). However, a characterisation of its exact switching profile and asymmetric error characteristics is pending.
Multiplexed single-photon source
The proposed novel multiplexed single-photon source exploits asymmetric configuration delays of switches while combining time- and space-multiplexing to improve the fidelity and excitation probability of photons. The components of the proposed photon source are available with current technology, can be manufactured at scale and are compatible with CMOS transistors as well as telecom frequencies (Jacques et al., Reference Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien and Laing2015; Xiong et al., Reference Xiong, Bell and Eggleton2016). Therefore the photon source presents itself as an ideal candidate for the demonstration of large-scale quantum computations with linear optics.
The proposed photon source consists of one or more SPDC sources, which are connected to the same pump laser, single- and double-padded switches, non-photon-number-resolving detectors, delays and logic elements for processing the detector signals, for example, CMOS transistors. Figure 3 shows the developed photon source with two SPDC sources that are connected via switches to an inner delay loop and the output. Each SPDC source is connected to a detector and to the inner loop via multiple switches that are configured at most twice. In each pump cycle of one SPDC source, a photon pair can be generated with probability p. When the attached detector indicates that one photon pair was generated at its corresponding SPDC source, one photon in the photon pair is absorbed by the detector and the other enters the inner loop by configuring the first switch that has not been used by the proposed source, yet. For instance, the first of the down-converted photons enters the inner loop by configuring switch S 1 in Figure 3, the second generated photon is entered by switch S 2, and so on. The delay element directly attached to each SPDC source delays a passing photon by
Figure 3. Developed photon source with two SPDC sources, seven double-padded switches that connect the sources to an inner delay loop and one single-padded switch S 8 that connects the inner loop to the output. figure 3(b)) is a more accurate representation of what figure 3(a)) would look like as an actual device. While figure 3(a)) demonstrates the developed design abstractly, figure 3(b)) examines the layout for two parallel sources, each with 16 switches, as it would appear if it were fabricated. figure 3(c)) serves as a key that maps the components in the schematic as depicted in figure 3(a)) to the components in an actual device depicted in figure 3(b)).
$$T=T_{d}+T_{c}+T_{sr},$$
where T d is the detector delay, T c is the classical processing delay and T sr is the previously defined switch configuration delay. After the photon has entered into the inner loop by a switch S i , the same switch S i is configured to forward newly generated photons along the outer loop to the next switch S i + 1 before the newly inserted photon traverses the inner loop once. This is facilitated by the inner loop delay T − ε, where T is the previously defined sum of delays and ε is the time a photon requires to traverse the inner loop once (without being delayed by ε). When another photon is generated, it will be forwarded to the next double-padded switch S i + 1 that is configured to switch the inner loop photons to the outer loop where the photon is discarded or output. The newly generated photon is switched into the inner loop by switch S i + 1 at the same time. Then, the same double-padded switch (S i + 1) is configured to keep the photon in the inner loop, and the next switch S i + 2 is used when a new photon is generated. This is repeated until all switches of a photon source are exhausted. The extension to multiple SPDC sources is straightforward: all detector signals are attached to classical processing, for example, CMOS gates, to select the generated photon of one SPDC source that is forwarded to the inner loop. After a fixed number of pump cycles, or when the component at the output of the proposed source requests, the photon from the inner loop is switched to the output. With this setup, each switch is reconfigured at most twice, which takes full advantage of asymmetric switch configuration delays.
At some point during the operation of the proposed single-photon source, a large number of switches have been configured which warrants a reset of the photon source. During the reset, the photon source cannot output any photon and the switches are reverted to their default configuration state, which requires a technology-dependent delay T sf . While the delay T sf may be large, the proposed single-photon source can be designed to alleviate it by reducing the rate at which a reset is necessary. The reset rate can be reduced by either connecting each SPDC source through additional switches or by using multiple single-photon sources. Both strategies extend the period after which a reset is necessary. If the detector delay T d is larger than the switch configuration delay T sr , detector multiplexing can be employed to reduce T d to T sr .
The pump frequency is given by T −1, where T represents the interval between pumping each probabilistic source and is bounded by the response time of the heralding detector, the classical processing time and the switching time (see Equation 2). However, after a full sequence of N pumping cycles is completed, the entire single-photon source must be reset. All thermal pads on all switches are deactivated and set to the default configuration. For this reset time, T sf ≫ T sr can be chosen to ensure that all switches have cleanly returned to the θ = 0 state without error.
As the multiplexed photon source must be reset periodically, the total repetition rate of the source can be much lower than the cycling time T −1 of the pumped source. The reset repetition rate t rmux −1 of the source is determined by
$$t_{{\rm rmux}}=N\times T+T_{sf}={\log (1-p_{N}) \over \log (1-p)}\times T+T_{sf}.$$
By prolonging t rmux the frequency of resets can be reduced and thus the total repetition rate of the source can be improved. The quantity t rmux can be increased by adding more switches to the inner loop that enable the routing of photons from SPDC sources to the inner loop or from the inner loop to the output.
Each photon source is attached to m switches that can be used to switch a generated photon into the inner loop of the racetrack. The number of attached switches m should be minimal to reduce the cost of manufacturing the proposed multiplexed photon source. At the same time, m must be large enough to allow a large portion of photons generated by the single-photon sources to be switched into the inner loop of the proposed multiplexed photon source. The error simulation performed in this work assumes that a sufficient number of switches are attached for each single-photon source. The minimum number of switches m can be derived by evaluating the inverse cumulative distribution function of the following binomial distribution
$$f(k,N,p)=Pr(X=k)=\left( {\matrix{ N \cr k \cr } } \right)p^{k}(1-p)^{N-k},$$
where N is the number of pump cycles, p is a given photon generation probability and f(k,n,p) is the probability of one photon source generating exactly k output photons in N pump cycles. If all k generated photons should be switched into the inner loop, the photon source requires m: = k switches. Hence, we can use the binomial distribution to determine an upper bound on the required number of switches such that each photon generated by a single-photon source has a large chance of getting switched into the inner loop.
We can compute the probability of generating at most k photons in N pump cycles, that is, the probability that a photon source requires m: = k switches by computing the cumulative distribution function of that binomial distribution
$$\widehat{\cal F}(k,N,p) = \sum\limits_{i = 0}^k {\left( {\matrix{ N \cr i \cr } } \right){p^i}{{(1 - p)}^{N - i}}}.$$
Finally, we can determine the minimum number m for a specified photon generation probability p and the number of pump cycles N as
$$m=\hat{\mathcal{F}}^{-1}(q,N,p),$$
where the inverse CDF
$\hat{\mathcal{F}}^{-1}$
returns the number m that is minimal such that
$$q \le \sum\limits_{i = 0}^m {\left( {\matrix{ N \cr i \cr } } \right){p^i}{{(1 - p)}^{N - i}}}, $$
with q being the probability of being able to switch all generated photons of one SPDC source during the N pump cycles into the inner loop of the racetrack. The inverse cumulative distribution function
$\hat{\mathcal{F}}^{-1}$
can be evaluated numerically (Virtanen et al., Reference Virtanen, Gommers, Oliphant, Haberland, Reddy, Cournapeau, Burovski, Peterson, Weckesser and Bright2020).
The data for Figures 4, 5 and 6 has been obtained via simulation: First, a random number r between zero and one is drawn for each considered SPDC source and pump cycle. Then, the pump cycle is determined at which the last photon in the multiplexed photon source has been generated. The pump cycle is determined by comparing each randomly drawn number r against the photon generation probability given for that simulation. In the last step, the number of pump cycles, the waveguide loss and the loop delay given for that simulation are considered to determine the probability of the most recently generated photon exiting the multiplexed photon source as described in the following section.
Figure 4. Photon output probability as a function of inner loop delay, number of pump cycles N and photon generation probability p with p ranging from 0.01% to 5% and
$N={\log (1.0-0.999) \over \log (1.0-p)}$
. The photon output probability p is increased by the value of 0.237% for each of the depicted curves.
Figure 5. Photon output probability for S ∈ {1, 5, 20, 50} photon sources (top left, top right, bottom left, bottom right) and a loop delay of T ∈ {1, 5, 20, 50, 100}ns at a waveguide loss of 0.024 dB/ns.
Figure 6. 3D plot of several configurations of 150 SPDC sources running for 150 pump cycles showing the achievable photon output probability as a colour from dark blue to yellow for an inner loop delay ranging from 1 ns to 100 ns, a waveguide loss ranging from 0.001 dB/ns to 24 dB/ns, and a photon generation probability of a single SPDC source ranging from 0.01% to 3%.
We show the photon output probability in Figure 4 as a function of inner loop delay T − ε, number of considered pump cycles N and photon generation probability p for one SPDC source attached to the outer loop. For a decreasing photon generation probability and increasing inner loop delay, the photon output probability diminishes exponentially as detailed in Figure 4. At a photon generation probability of 5% and an inner loop delay of 9ns the photon output probability is roughly at 66% when performing 134 pump cycles. This exponential decrease in photon output probability can be mitigated by introducing additional SPDC sources to the inner loop.
In Figure 5, the photon output probability of the proposed single-photon source is shown as a function of attached SPDC sources and pump cycles. The number of pump cycles is shown on the x-axis and the probability of photon output is given on the y-axis. The different lines indicate an inner loop delay ranging from 1 ns to 100 ns.
For the lowest considered inner loop delay of 1ns, the photon output probability converges to 100% for 5, 20 and 50 SPDC sources. At one SPDC source, it reaches up to 95%. For more than one considered SPDC source, a sharp rise in photon output probability is observable in the region of one to ten pump cycles. At higher inner loop delays, the photon loss increases and yields a diminished photon output probability. At 5 ns inner loop delay the photon loss leads to a photon output probability of less than 90% for one source, whereas the photon output probability converges to 23% for an inner loop delay of 100 ns. At 5, 2, and 50 SPDC sources this effect is reduced; a 100 ns inner loop delay leads to a photon output probability of 62% for 5 SPDC sources, 83% for 20 SPDC sources and slightly over 88% for 50 SPDC sources. At an increasingly larger number of SPDC sources, the depicted curves become less smooth. This is because of the probabilistic nature of our simulation and a finite sampling of ten thousand repetitions per data point.
Figure 6 shows a 3D plot where the inner loop delay on the x-axis, the waveguide loss per nanosecond on the y-axis and the photon generation probability of the SPDC sources on the z-axis is varied. The colour encodes the photon output probability of a racetrack design subject to a value assignment of the three parameters as given on their respective axes. The number of SPDC sources and number of pump cycles is set to 150 for all data points. It can be observed that a higher inner loop delay can mostly be compensated by a higher photon generation probability of the SPDC sources for an inner loop delay of less than 40 ns. Above an inner loop delay of 40 ns, the photon output probability converges to a lower value regardless of the photon generation probability. The top face of the cube shows that the waveguide loss, inner loop delay and photon generation probability must be in a very restricted region: to allow for photon output probabilities of more than 84%, the waveguide loss must be less than 5db/ns and the inner loop delay must be less than 10 ns at a photon generation probability of 3%. The photon output probability converges quickly to zero for an increasing waveguide loss and inner loop delay.
Error modelling
The inner loop delay T incurs photon loss. We modelled this photon loss by the power law, that is
$$e^{-T\cdot I\cdot L},$$
gives the probability that a photon is not lost while traversing the inner loop I times at a waveguide loss of L. The range of values investigated in the parameters of the results figures are justified by values reported in experimental setups of previous art (Lee et al., Reference Lee, Chen, Li, Painter and Vahala2012; Bauters et al., Reference Bauters, Davenport, Heck, Doylend, Arnold, Fang and Bowers2013; Vigliar et al., Reference Vigliar, Paesani, Ding, Adcock, Wang, Morley-Short, Bacco, Oxenløwe, Thompson, Rarity and Laing2021; Lita et al. Reference Lita Adriana, Miller Aaron and SaeWoo2008; Kaneda and Kwiat, Reference Kaneda and Kwiat2019; Meyer-Scott et al., Reference Meyer-Scott, Silberhorn and Migdall2020; Sabouri et al., Reference Sabouri, Angel Mendoza Velasco, Catuneanu, Namdari and Jamshidi2021; Liu et al., Reference Liu, Tian, Lu and Feng2021, Reference Lu, Zhou, Gühne, Gao, Zhang, Yuan, Goebel, Yang and Pan2022). The work in Bauters et al. (Reference Bauters, Davenport, Heck, Doylend, Arnold, Fang and Bowers2013) and Lee et al. (Reference Lee, Chen, Li, Painter and Vahala2012) reports a waveguide loss of 0.024 dB/ns, while Vigliar et al., (Reference Vigliar, Paesani, Ding, Adcock, Wang, Morley-Short, Bacco, Oxenløwe, Thompson, Rarity and Laing2021) reports a waveguide loss of 16 dB/ns. We have chosen a detector efficiency of 90% in all of our simulations; in (Vigliar et al., Reference Vigliar, Paesani, Ding, Adcock, Wang, Morley-Short, Bacco, Oxenløwe, Thompson, Rarity and Laing2021) a detection efficiency of 78% ±5% and in (Lita et al., Reference Lita Adriana, Miller Aaron and SaeWoo2008) a detector efficiency of 95% is reported. Works in Kaneda and Kwiat (Reference Kaneda and Kwiat2019) and Meyer-Scott et al. (Reference Meyer-Scott, Silberhorn and Migdall2020) report a photon generation probability of 1%, whereas the work in Vigliar et al. (Reference Vigliar, Paesani, Ding, Adcock, Wang, Morley-Short, Bacco, Oxenløwe, Thompson, Rarity and Laing2021) reports a photon generation probability of 3%. As we are looking at the single-photon source in the context of fault-tolerant optical quantum computing, multi-photon contamination “purity” has to be bounded as the error correction in fault-tolerant optical quantum computing is not designed to handle multi-photon events. The probability of success also allows us to bound the amplitude of the multi-photon terms in Equation 1. At approximately a 1% success probability (i.e. ζ 2 ≈ 1% in Equation 1), the next higher order term occurs with a probability of ζ 4 ≈ 10−4, which is comfortably below threshold for surface-code and Raussendorf lattice-based error correction schemes (Fowler et al., Reference Fowler, Mariantoni, Martinis and Cleland2012).
Works in Liu et al. (Reference Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo2022); Liu et al. (Reference Liu, Tian, Lu and Feng2021); Sabouri et al. (Reference Sabouri, Angel Mendoza Velasco, Catuneanu, Namdari and Jamshidi2021) report a switch configuration delay of 50 ns for the first and second configuration (T sr ) and a delay of 950ns for the cooldown required for a third configuration. While thermo-optic switches can be comparatively slow, they also exhibit extremely low-loss rates as soon as they are set. A larger number of switches in the inner loop, however, requires a larger waveguide length and thus leads to larger loss rates due to the intrinsic loss of photons. Photon detectors are expected to have a photon absorption delay of several picoseconds, which is negligible. However, photon detectors are also assumed to have a relatively large dead (or: reset) time that is on the order of the switch-off time. Classical processing is expected to be handled with CMOS gates, which incur a negligible delay of a few hundred picoseconds.
Conclusion
We have presented a multiplexed single-photon source that combines space- and time-multiplexing to amplify the photon output probability. Depending on the underlying technology parameters, the proposed single-photon source can be tuned to use more time steps (pump cycles) or more SPDC sources. The proposed single-photon source can be manufactured at scale with currently available technology. The source can exploit asymmetric configuration delays by using thermo-optic switches with two heat pads at most twice during its operation. Simulations have shown that the proposed source can reach a photon output probability of almost 87% at an inner loop delay of 20 ns and five single-photon sources attached to the inner loop. This work reinforces the fact that loss is by far the most dominant source of error that needs to be overcome before any scheme to improve probabilistic sources of photons can be realised.
Our goal was to examine a redesign that we initially hypothesised could lead to potentially ultrafast operations with comparatively low speed and low-loss heating pad modulators. While we did show, theoretically, an improvement over naive approaches to optical multiplexing, the ultimate goal of producing a 1 GHz pseudo-deterministic source under realistic loss models was not achieved. Still, the ideas presented in this work may prove useful in multiplexed photon source design in the future, when new materials (especially for low-loss waveguides and fast electro-optical modulation are demonstrated).
Data availability statement
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Acknowledgements
We thank Stefanie Barz and Jeldrik Huster for helpful discussions.
Author contribution
SD and IP conceived and designed the study. SD, CM and SB conducted data gathering. SD, SB performed statistical analyses. All authors wrote the article.
Financial support
This work was partially funded by the Carl Zeiss foundation.
Competing interests
The authors declare no conflicts of interest.
Ethics statement
Ethical approval and consent are not relevant to this article type.
Open access
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