Systems and methods for solving tacit coordination problems using quantum telepathy

Abstract

Systems and methods for tacit coordination of decisions. In one embodiment, a system includes first and second measurement devices that are each coupled to respective quantum memory devices. A heralded entanglement measurement device photonically connecting the quantum memory devices via respective quantum channels is configured to receive photons emitted from the respective quantum memory devices via the quantum channels and to perform measurements on the photons to determine whether the quantum memory devices are entangled. The heralded entanglement device transmits a heralding signal to the first and second memory devices indicating whether the memory devices have been successfully entangled. If they are successfully entangled, the memory devices are configured to make tacitly coordinated decisions using local information and thereby obtain a quantum advantage. Other embodiments of systems are disclosed as are methods for optimizing and tuning the systems and for using the systems in various applications including high frequency trading, distributed computing, and computer architecture.

Description

SYSTEMS AND METHODS FOR SOLVING TACIT COORDINATION PROBLEMS USING QUANTUM TELEPATHYPRIORITY CLAIM

[0001] This application claims priority to U. S. Provisional App. Ser. No. 63 / 677,887, filed July 31, 2024.BACKGROUND OF INVENTION

[0002] Field of Invention

[0003] The present invention relates to systems and methods for solving tacit coordination problems using quantum telepathy.

[0004] Brief Description of Related Art

[0005] Quantum telepathy is the phenomenon where two non-communicating parties can exhibit correlated behaviors that are impossible to achieve using classical mechanics. This is also known as Bell inequality violation and is made possible by quantum entanglement. In general, it is often problematic to coordinate decisions given a set of observations without being able to communicate. This inability is actually quite prevalent in the modern era, where the decision-making timescales of computer processors are so short that the speed of light delay is actually quite appreciable in comparison. One example is in the field of high-frequency trading (HFT), where trades are made at microsecond timescales, but the speed of light delay between different stock exchanges can range from the order of 100 microseconds to 10 milliseconds.

[0006] Quantum telepathy is not a just topic of fundamental physics or philosophy. Quantum telepathy is a technology with real-world applications. See, e.g., Szegedy et al., U. S. Pat. Nos. 11,676,104 B2 and 12,165,108 B2. To better elucidate how Bell inequalities can appear in real-world settings, the aforementioned patents disclosed a new concept called “coordinating decisions between non-communicating parties” (CDNP) problems, which are hereinafter referred to herein more simply as tacit coordination (TC) problems. This terminology is being utilized because some generalizations and slight deviations from the usual theory of Bell inequalities and the equivalent computer science concept of nonlocal games are made herein. In short, aTC problem arises when there are multiple (i.e., two or more) non-communicating parties who each make a local observation and a local decision. The aim is to optimize a global “utility” of their collective decisions based on their collective observations and thereby obtain a quantum advantage over what could be achieved using classical mechanics.BRIEF SUMMARY OF THE INVENTION

[0007] According to a first embodiment, a system for tacit coordination of decisions, the system comprising: (a) a device for generating entangled photon pairs having a measurable state; (b) a first measurement device; (c) a second measurement device; (d) a first quantum channel photonically connecting the device for generating the entangled photon pairs and the first measurement device; and (d) a second quantum channel photonically connecting the device for generating the entangled photon pairs and the second measurement device. The first measurement device is configured to measure the measurable state of first photons of the entangled photon pairs generated by the device for generating the entangled photon pairs that are successfully received by the first measurement device based in part on a measurement setting determined using first local information and thereby obtain a first measurement result. Similarly, the second measurement device is configured to measure the measurable state of second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs that are successfully received by the second measurement device based in part on a measurement setting determined using second local information and thereby obtain a second measurement result. The first measurement device does not know whether the second measurement device successfully received its entangled photon, and vice versa. To make tacitly coordinated decisions that achieve a quantum advantage, the first measurement device is configured to make a first local decision. The first local decision is made using a predetermined quantum strategy based on the first measurement result when the first photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are successfully received by the first measurement device, but is made using a predetermined deterministic strategy when the first photons of the entangled photonpairs generated by the device for generating the entangled photon pairs are not successfully received by the first measurement device due to photon loss in the first quantum channel. Similarly, the second measurement device is configured to make a second local decision, which is made using the predetermined quantum strategy based on the second measurement result when the second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are successfully received by the second measurement device, and is made using the predetermined deterministic strategy when the second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are not successfully received by the second measurement device due to photon loss in the second quantum channel. The predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account.

[0008] In one application, the first measurement device can be located in a colocated server of a first stock exchange, and the first local information comprises a real-time market data feed from the first stock exchange. The second measurement device can be located in a colocated server of a second stock exchange, and the second local information comprises a real-time market data feed from the second stock exchange. The tacitly coordinated decisions can be trade decisions executed by the servers at the respective first and / or second stock exchanges.

[0009] The first and / or second quantum channels can be selected from the group consisting of vacuum beam guides, optical fibers, free space, or optical waveguides.

[0010] The measurable state of the entangled photon pair can be encoded in their polarization degrees of freedom. When this is the case, the first measurement device can comprise a waveplate and a polarizing beam splitter that routes photons with different polarization to different respective photon detectors. The waveplate is preferably tunable based on the first local information. The tacitly coordinated decision made by the first measurement device are preferably based upon which of the different respective photon detectors measured the measurable state of an entangled photon of the entangled photon pair.

[0011] The measurable state of the entangled photon pair can be encoded in their spatial modes. When this is the case, the first measurement device can be configuredto convert spatial mode encoding to polarization, and the first measurement device can comprise a waveplate and a polarizing beam splitter that routes photons with different polarization to different respective photon detectors. The device for generating the series of entangled photon pairs can be configured to perform spatial mode encoding with a fixed linear optical module.

[0012] It will be appreciated that the device for generating the series of entangled photon pairs can be configured to perform encoding of cluster states via electrons emitting photons, or can be configured to perform bosonic quantum encoding that is robust against photon loss errors. The system can be configured to use bosonic quantum error correction to suppress photon loss errors of up to 50% loss to deterministically generate entanglement. Preferably, quantum error correction is performed before the measurement to suppress loss error. The device for generating the entangled photon pair can comprise an optically pumped nonlinear crystal, and can be configured to perform time-bin encoding with fast linear optics.

[0013] In another disclosed application, the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first server and a second server, respectively, that are spaced apart from each other such that their decisions have to be made after their observations within a time less than their communication latency.

[0014] In another disclosed application, the system further comprises a preprocessor that divides a data stream into a first fraction that comprises the first local information and a second fraction that comprises the second local information.

[0015] In another disclosed application, the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first processor and a second processor, respectively, at a time scale less than their communication latency. The first processor and the second processor can have a shared memory and the system minimizes address collisions.

[0016] According to a second embodiment, a system for making tacitly coordinated decisions comprises: (a) a first measurement device; (b) a second measurement device; (c) a first quantum memory device coupled to the first measurement device; (d) a second quantum memory device coupled to the second measurement device; (e) aheralded entanglement measurement device; (f) a first quantum channel photonically connected between the heralded entanglement measurement device and the first quantum memory device; and (g) a second quantum channel photonically connected between the heralded entanglement measurement device and the second quantum memory device. The first and the second quantum memory devices are each configured to repeatedly emit photons toward the heralded entanglement device via the first and second quantum channels. The heralded entanglement device is configured to detect photon loss errors, and is configured to transmit heralding signals to the first and second quantum memory devices, respectively, indicating when the first and second quantum memory devices have been successfully entangled. The first measurement device is configured to perform a measurement of the first quantum memory device based in part on a measurement setting determined using first local information, and the second measurement device is configured to perform a measurement on the second quantum memory device based in part of a measurement setting determined using second local information. The first and second measurement devices are configured to make the tacitly coordinated decisions based on the measurements made by first and second measurement devices, thereby obtaining a quantum advantage. Preferably, the first and second measurement devices are configured to make the tacitly coordinated decisions using a predetermined optimal quantum strategy when they determine that there are sufficient entangled memory devices to do so, and are configured to make the tacitly coordinated decisions using a predetermined optimal deterministic strategy when they determine that there are insufficient entangled memory devices to do so.

[0017] To suppress depolarizing errors, the system is preferably configured to use an entanglement distillation procedure, which uses multiple Bell pairs and local operation and classical communication to distill Bell pairs with higher fidelity. The distilled Bell pairs can be used for teleported gates to obtain logical Bell pairs with quantum error correction protection.

[0018] The system can further comprise a first quantum transducer coupled to the first of the pair quantum memory devices and a second quantum transducer coupled to the second of the pair quantum memory devices. The first and second quantum transducers are configured to convert the frequencies of the photons emitted by theentangled quantum memory devices, while preserving entanglement, for transmission through the first and second quantum channels, respectively.

[0019] In one disclosed application, the first measurement device is located in a colocated server of a first stock exchange, and the first local information comprises a real-time market data feed from the first stock exchange. The second measurement device can be located in a colocated server of a second stock exchange, and the second local information can comprise a real-time market data feed from the second stock exchange. In such an application, the tacitly coordinated decisions can be trade decisions executed by the servers at the respective first and / or second stock exchanges.

[0020] As in the first embodiment, the first and / or second quantum channels can be selected from the group consisting of vacuum beam guides, optical fibers, free space, or optical waveguides.

[0021] In a preferred embodiment, the system comprises first and second lasers configured to drive the first and the second of the pair of quantum memory devices, respectively, to emit the photons that the heralded entanglement device receives via the first and second quantum channels. The system also preferably further comprises first and second microwave drives configured to perform quantum gate operations on the first and second of the pair of quantum memory devices based on local information when the heralding signal indicates that the quantum memory devices have been successfully entangled.

[0022] In another preferred embodiment, the system further comprises first and second measurement lasers directed at the first and second of the pair of quantum memory devices that generate state dependent fluorescence which the first and second measurement devices are configured to detect to make the tacitly coordinated decisions. In such an embodiment, the first and second measurement devices can each comprise a camera for detecting the state dependent fluorescence.

[0023] The pair of quantum memory devices preferably comprise at least one of: (a) atoms; (b) ions; (c) superconducting qubits; (d) nitrogen vacancy centers; (e) nuclear / electron spins; and (f) microwave cavities.

[0024] Also disclosed herein is a method for making tacitly coordinated decisions given first and second local information, respectively. The method comprises: (a) emitting a pair of photons, wherein the pair of photons includes a first photon emitted from a first quantum memory and a second photon emitted from a second quantum memory; (b) measuring the pair of emitted photons to determine whether the first and second quantum memories are entangled; (c) when the first and second quantum memories are determined to be entangled, sending heralding signals to the first and second quantum memories that indicates successful or unsuccessful entanglement; (d) storing the successful entanglement in the first and second quantum memories, respectively; (e) performing quantum operations on the first and second quantum memories given the first and second local information; (f) consuming the stored successful entanglement by measuring the first and second quantum memories; and (g) making the tacitly coordinated decisions based on the measurements of the first and second quantum memories to obtain a quantum advantage.

[0025] In one application of the method, the first local information comprises a realtime market data feed from a first stock exchange, and the second local information comprises a real-time market data feed from a second stock exchange. In accordance with the method, the tacitly coordinated decisions can be made faster than latency in communication between the first stock exchange and the second stock exchange.Preferably, the real-time market data feed from the first stock exchange and the realtime market data feed from the second stock exchange include data that indicates a statistical relationship between a plurality of financial instruments. Also preferably, the tacitly coordinated decisions are trading decisions. In some instances, the trading decisions are used for hedging.

[0026] In another preferred application of the method, the tacitly coordinated decisions are computations made by a plurality of servers performing distributed computing. In another example, the tacitly coordinated decisions can be instructions executed by separate processors in a computer. In accordance with the method, the separate processors can read from a shared memory and the tacitly coordinated decisions minimize address collisions. In such instance, the first and second localinformation can comprise addresses in the shared memory that the separate processors are to query.

[0027] Preferably, a determination is made whether depolarizing noise from memoryphoton entanglement, imperfect photon detection, decoherence of the first and second memory devices during photon traversal and / or measurement of the first and second memory devices exceeds a predetermined robustness value such that the tacitly coordinated decisions made at the predefined frequency obtain the quantum advantage.

[0028] A second method for making tacitly coordination of decisions that achieve a quantum advantage, the method comprises: (a) generating entangled photon pairs having a measurable state; (b) transmitting, via first and second quantum channels, respectively, first photons of the entangled photon pairs to a first measurement device and second photons of the entangled photon pairs to a second measurement device; (c) measuring, using the first measurement device, the measurable state of the first photons of the entangled photon pairs that are successfully received by the first measurement device based in part on a measurement setting determined using first local information to thereby obtain a first measurement result; (d) measuring, using the second measurement device, the measurable state of the second photons of the entangled photon pairs that are successfully received by the second measurement device based in part on a measurement setting determined using second local information to thereby obtain a second measurement result; and (e) to achieve the quantum advantage, making a first local decision via the first measurement device using a predetermined quantum strategy based on the first measurement result when the first photons of the entangled photon pairs are successfully received by the first measurement device and using a predetermined deterministic strategy when the first photons of the entangled photon pairs are not successfully received by the first measurement device due to photon loss in the first quantum channel, and making a second local decision via the second measurement device using the predetermined quantum strategy based on the second measurement result when the second photons of the entangled photon pairs are successfully received by the second measurement device and using the predetermined deterministic strategy when the second photons of the entangled photon pairs are not successfully received by the second measurementdevice due to photon loss in the first quantum channel. The predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account.

[0029] The foregoing and other features of the invention are hereinafter more fully described below, the following description setting forth in detail certain illustrative embodiments of the invention, these being indicative, however, of but a few of the various ways in which the principles of the present invention may be employed.BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The above and other characteristics and advantages of the invention will be better understood through the following illustrative and non-limitative detailed description of preferred embodiments thereof, with reference to the appended drawings, wherein:FIG. 1 is a schematic illustration of a tacit coordination problem involving two parties;FIG. 2 illustrates a high-frequency trading scenario where trade decisions are being made by servers receiving stock price updates from markets that are separated from each other by a distance;FIG. 3A shows the quantum advantage for an exemplary hedging problem as a function of p and 3, where each ranges from 0 to 1 in increments of 0.1; FIG. 3B shows quantum advantage for the exemplary hedging problem as a function of for p = 0.5;FIG. 3C shows the quantum advantage for the exemplary hedging problem as a function of p for 3 = 0.4;FIG. 4 is a schematic illustration of a system for tacit coordination of decisions according to a first embodiment of the invention;FIG. 5 shows greater detail of the subject matter appearing within the dashed box in FIG. 4;FIG. 6 is a schematic illustration of one specific implementation of a system according to FIG. 4;FIG. 7 is schematic drawing of two CPU's making read accesses to two copies of memory;FIG. 8 is an exemplary utility matrix;FIG. 9A shows 1 - p* values computed for an exemplary hedging problem with p and p taking values between 0 and 1 with increments of 0.1;FIG. 9B shows p* as a function of p for the exemplary hedging problem when p = 0.5;FIG. 9C shows p* as a function of p for the exemplary hedging problem when p = 0.4;FIG. 10 is a flowchart showing steps of a method that can be implemented using a system according to FIG. 4;FIG. 11A shows robustness values computed for an exemplary hedging problem with p and 3 taking values between 0 and 1 with increments of 0.1;FIG. 11 B shows robustness as a function of 13 when p = 0.5;FIG. 11C shows robustness as a function of p when 3 = 0.4;FIG. 12A shows the quantum advantage for the exemplary hedging problem as a function of p and (3, where each ranges from 0 to 1 in increments of 0.1, for various levels of depolarizing noise;FIG. 12B shows the quantum advantage as the noise level increases;FIG. 12C shows the quantum advantage as the noise level increases further; FIG. 13 is a schematic illustration of a system for tacit coordination of decisions according to a second embodiment of the invention;FIG. 14 is a flowchart showing steps of a method that can be implemented using a system according to FIG. 13; andFIG. 15 shows a theorem by Tsirelson that illustrates a central result for XOR problems with two parties.DETAILED DESCRIPTION OF THE INVENTION

[0031] The following description and accompanying drawing figures describe certain embodiments by way of illustration only. One skilled in the art will readily recognize from the following description that alternative embodiments of the structures andmethods illustrated herein may be employed without departing from the principles described herein. Reference will now be made to several embodiments, examples of which are illustrated in the accompanying figures.

[0032] Systems according to the invention solve TC problems by achieving coordination using quantum telepathy, a phenomenon based on quantum entanglement. The described embodiments enable distribution of entanglement over long distances via photons. The disclosed embodiments include photonic implementations of quantum telepathy schemes that manage loss and other issues.

[0033] Quantum telepathy enables the coordination between multiple parties without communication. The disclosed embodiments include methods for physical implementations of quantum telepathy protocols including techniques for photonic distribution of entanglement. The described systems are suitable for various applications. Applications relating to different distance regimes may be implemented using different corresponding physical media suitable for photon transmission at those distances. The disclosed embodiments include methods for physical encodings of entangled quantum states into photons, and embodiments in which there exists a quantum memory at the different end nodes. The disclosed embodiments furthermore include methods for making tacitly coordinated decisions and methods for determining whether quantum entanglement can provide a quantum advantage for specific tacit coordination problems with relevant physical parameters such as loss rate, background noise levels, and photon source power. Some embodiments include a numerical optimizer capable of determining the quantum advantage if it exists.

[0034] Section 1: Introduction

[0035] Quantum entanglement is a unique testament to the strangeness of quantum mechanics. As stated in the celebrated result of John Bell, no local hidden variable theory can reproduce the correlated behaviors of entangled particles. His prediction, which is known as Bell inequality violation, and its subsequent experimental verification, completely overturned traditional assumptions of how nature works and made mathematically explicit the departure quantum mechanics makes from the familiar classical world. This phenomenon was also described by Einstein, Podolsky, and Rosen as “spooky action at a distance”: although they are not communicating,entangled particles can display behaviors that seem inconceivable and even uncanny. Hence Bell inequality violation is also known as quantum telepathy. How this is possible can be tied to the particularly counterintuitive nature of quantum entanglement. Stated plainly, just as relativity defies the ancient intuition that time is absolute, quantum entanglement defies the ancient intuition that a description of a combination of objects is the combination of their descriptions. Borrowing the words of Aristotle, for a system of entangled particles, “the whole is greater than the sum of its parts” in an information-theoretic sense.

[0036] Quantum telepathy is not a just topic of fundamental physics or philosophy. Quantum telepathy is a technology with real-world applications. To better elucidate how Bell inequalities can appear in real-world settings, the patents propose a new concept called “coordinating decisions between non-communicating parties” (CDNP) problems, which we will henceforth refer to more simply as tacit coordination (TC) problems. We also introduce this terminology because we will make some generalizations and slight deviations from the usual theory of Bell inequalities and the equivalent computer science concept of nonlocal games. In short, a TC problem arises when there are multiple non-communicating parties who each make a local observation and a local decision. The aim is to optimize a global “utility” of their collective decisions based on their collective observations. Note that for consistency, we will use this terminology throughout this specification. One very natural real-world setting where TC problems can appear is high-frequency trading (HFT). In this case, each party is a colocated server at a stock exchange engaged in HFT for the same firm. Each server has access to local information, such as stock price fluctuations at their respective exchange. Now, these stock exchanges are spatially separated by distances ranging from tens of kilometers to thousands of kilometers. The speed of light delay thus ranges from hundreds of microseconds to tens of milliseconds. However, modern HFT is conducted on timescales as short as microseconds and in the future may even shorten to nanoseconds. No matter how much effort is put into shortening latencies, with these numbers it is physically impossible for different servers to communicate before making a trade decision. However, there could be a nontrivial globally optimal pair of trades givena pair of observations to minimize risk or maximize expected returns. See Section 3 for an example. Thus, HFT is naturally a TC problem.

[0037] Now, we emphasize Bell inequality violation / s irrefutable proof of a quantum advantage. It does not require any complexity-theoretic assumptions such as BQP + BPP and is actually a straightforward mathematical argument. In fact, Bell inequality violation is so convincing that it can be used to prove quantum advantage in other settings, such as shallow quantum circuits. Furthermore, to achieve a quantum advantage we do not necessarily need complicated schemes such as fault-tolerant quantum computing. For example, violating the CHSH inequality only requires two physical qubits and a single-qubit gate prior to each measurement. Indeed, the number of qubits and fidelities necessary to violate the inequality have already been achieved half a century ago. Even for Bell inequalities that require high-dimensional quantum systems to achieve the maximum violation, it is sufficient to find (if it exists) a lowdimensional scheme that also violates the inequality, just not maximally, to attain a quantum advantage.

[0038] TC problems are actually quite prevalent in the real world and that a quantum advantage can be attained with currently available or near-future technologies. The core insight is that speed of light delays for distances on the order of 10 km is already 10 to 100 ps, which is a relatively long period of time in our modern world satiated with classical processors that have GHz clock speeds. In particular, we conduct a case study of a concrete HFT scenario that gives rise to a generalization of the CHSH game and assess different possible physical implementations of quantum telepathy schemes. In Section 2, we first give an informal definition of TC problems. Then, in Section 3, we present the concrete HFT scenario and compute what quantum advantages can be attained. In Section 4, we go into the detailed physical implementations of quantum telepathy schemes. In particular, we evaluate their ability to achieve a quantum advantage for the said HFT scenario, considering practical issues such as photon loss, entanglement generation rates, and robustness to noise. We end with a discussion in Section 5, where we consider other settings where TC problems may arise such as distributed computing and computer architecture. We also provide technical definitions and prove facts about TC problems that we use in our analyses. In addition, we presenta numerical optimizer for computing the quantum advantage attainable for general TC problems.

[0039] Section 2: Definition of TC Problems

[0040] We give an informal definition of TC problems. For a technical exposition, see Appendix A. A TC problem involves multiple parties that each make an observation followed by a decision so as to maximize a global utility. FIG. 1 schematically illustrates a TC problem. For simplicity, we will assume there are two parties: 1 and 2. Party 1 makes observation 01 and subsequent decision di while Party 2 makes observation 02 and subsequent decision d2.

[0041] A utility array describes what is the utility of a set of decisions given a set of observations. For simplicity, suppose each party can only have two possible observations {o, 0} and two possible decisions {d, d. The utility array can be written in matrix form:

[0042] where the rows are labeled by possible pairs of observations and the columns are labeled by possible pairs of decisions. The matrix element is then the utility of that pair of decisions given that pair of observations. The parties are to each make a decision given their observation that maximizes the expected value of the utility, given an input distribution, that is, a probability distribution over the observations. The catch is that the parties are cannot communicate during this decision-making process. The reason for this could be various:• The parties may be spatially separated and have to make decisions faster than the communication latency or even the speed of light delay.• The parties may have lost their communication link due to a technical fault or other problem.

[0043] The former case can be quite prevalent in modern settings: take the parties to be computers that perform tasks with GHz clock speeds. As long as the spatialseparation between the computers is more than 30 cm, communication is physically impossible.

[0044] Without communication, how do the parties maximize their utilities? If they’re smart, the parties would have agreed on a strategy beforehand given knowledge of the utility array. The implementation of the strategy leads in general to a probability distribution of decisions conditioned on observations, which can also be expressed as a matrix:

[0045] Because these are probabilities, this is a right stochastic matrix, i.e. the rows sum to 1. We will call this conditional probability distribution the behavior of the parties. The expected utility is then given by multiplying these probabilities with the entries of the utility array and some input distribution po(o).

[0046] In a deterministic strategy, each individual party simply makes a decision based on their local observation:1. Before starting, the parties prepare respective functions fi, f2: {o, o'} — {d, d based on the utility matrix.2. After starting, the parties compute their respective functions by taking their respective local observation as input and use the corresponding output as their decision.

[0047] In general, a classical strategy, also known as a local hidden variable theory, can involve randomness: both local randomness for each party and shared randomness establishing correlations between different parties. However, the expected utility of any classical strategy is always a convex combination of the expected utilities of deterministic strategies. Hence, to find the maximum possible expected utility function of all classical strategies, it is sufficient to only consider deterministic strategies.

[0048] Now, the phenomenon of Bell inequality violation in the language of TC problems is this: for certain TC problems, we can devise strategies using quantummechanics that achieve higher expected utilities than that of any strategy we can devise classically. Such quantum strategies are of the following form:1. Before starting, the parties share entangled particles. The precise state of the entangled particles is specifically designed for the utility matrix.2. After starting, the parties apply a quantum measurement to their particle, the type of measurement being based on their observation.3. The parties make decisions based on the measurement result.

[0049] Note here that no communication is conducted at any step. However, the parties need to be able to share entangled particles.

[0050] In general, we will be interested in the quantum value and classical value, which are defined as the expected utilities optimized over quantum and classical strategies, respectively. We will refer to the difference between the quantum and classical value as the gap, or quantum advantage. For detailed technical definitions of the concepts in this section, see Appendix A.

[0051] Section 3: Latency TC Problems

[0052] We currently live in a world where the speed of light delay between different parties can be appreciable compared to the timescales in which decisions need to be made. Classical processors have GHz clock speeds, while light in vacuum can only travel 30 cm in 1 ns. Hence, TC problems can easily arise in real-world scenarios where communication is not possible due to latency. In general, we give the following criteria for a latency TC problem:1. Multiple parties are involved that each make local observations and decisions.2. There is a global utility associated with a set of decisions given a set of observations.3. The parties do not have enough time to communicate their observations with each other before having to make a decision.

[0053] The last criterion could be due to fundamental physics constraints such as speed of light (in vacuum) delay or other factors such as having only an optical fiber connection through which light has to travel farther than the displacement between the parties and travels more slowly than in vacuum.

[0054] Section 3.1: High Frequency Trading

[0055] In this subsection we will conduct a detailed case study involving a high frequency trading scenario where quantum entanglement can provide an advantage. We will attempt to make the scenario as detailed as possible, but some simplifications will be necessary due to lack of real market data and to keep things conceptually straightforward. The TC problem we present should therefore be interpreted as a toy model. In general, to obtain concrete quantitative predictions of quantum advantage in practical settings, real historical HFT data is needed.

[0056] Consider the HFT setup in FIG. 2. Suppose a market maker named Zhuo operates in NYSE and NASDAQ, where he respectively trades correlated stocks X and Y. In layman’s terms, Zhuo is a supplier: he provides trades (both buying and selling) for a stock X listed in NYSE and a stock Y listed in NASDAQ near current market prices so that trading can proceed. Zhuo has two colocated servers, one at each exchange. Each server receives market data from its local exchange (observations) and modifies its HFT algorithm accordingly (decision). Thus the first criterion for latency TC problems is satisfied. Since the stocks are correlated, the pair of decisions given the observations will determine how favorable Zhuo’s position is given the market information at both exchanges (utility). Thus the second criterion is fulfilled. Furthermore, the data centers of these two stock exchanges are around 35 miles, or 56.3 km, apart which means even a line of sight connection in vacuum would entail a speed of light delay of at least about 188 ps. However, HFT is sometimes conducted on timescales as short as microseconds, which means in general the two colocated servers do not have enough time to communicate before they have to make a trade decision. Thus the third criterion is satisfied.

[0057] We now provide the full details of this particular TC problem. As Zhuo is a market maker, each server conventionally issues orders in pairs where one order is a bid and the other an ask. However, the pair of orders is issued sequentially and the first order is in general more likely to be filled. This is due to the first order arriving earlier and having being made with slightly more up-to-date information. Thus, if a bid order is always issued first in the pair, then there will be a slight bias toward successful bid orders over ask orders. Which order to issue first corresponds to the decision to bemade in the TC problem for each server. It will be appreciated that, in general, this will depend on the order types available at the exchanges. For example, some order types send both the ask and bid orders in one message, but even then the server has to sequentially decide on their respective price values. A possible exception to this is if the order type allows for specifying a spread.

[0058] Now, assume the price movements of stock X and Y are usually positively correlated. In this case, Zhuo should let the servers make opposite decisions in order to hedge, that is, to reduce risk. That is, if one server decides to ask first, the other should bid first. This way Zhuo’s portfolio will effectively have a nice tradeoff between two positively correlated stocks, thereby reducing the variance of his position. However, each server will also look for a technical indicator that stock X and Y are now negatively correlated. The precise nature of such an indicator would need to be empirically determined in real-world settings. This corresponds to the observation in the TC problem that each server makes. In this case, Zhuo should let the two servers make the same decision, again for hedging. Now, both exchanges are looking for this indicator. If both see the indicator, then Zhuo should definitely let the servers make the same decision. If only one sees an indicator, there can be slight preference for making the same decision over opposite decisions or vice versa.

[0059] At a high level, we see that in this HFT scenario the utility (favorability of Zhuo’s position) is only dependent on whether the servers make the same decision or different decisions. Hence, it is an instance of an XOR game. Here, the XOR of the decisions of the two servers has a natural trading interpretation as hedging for positively or negatively correlated stocks. We next wish to explicitly write out the utility array. Assuming that only one indicator of potential negative correlation is not sufficient to prefer making the same decision over opposite decisions, the utility array is actually that of the opposite winning conditions of the CHSH game:

[0060] where the observations N, I correspond to “No indicator” versus “Indicator” of negative correlation, and the decisions A, B correspond to “Ask-first” or “Bid-first” order issuing, respectively. We will choose the order of parties as the NYSE server followed by the NASDAQ server. This establishes that TC problem such as the CHSH game can manifest in real-world scenarios. For simplicity, we assume that the figure of merit is indeed the expected utility. This is sensible for special cases. Now, to quantitatively evaluate a quantum advantage, we also need to consider the input distribution.Suppose the input distribution is independent Bernoulli distributed with parameter p for both servers, where with probability p the indicator is observed (I). We analytically solve this TC problem in Appendix A1, where we find that a quantum advantage exists if:

[0061] We can also consider a natural extension where one server observing the indicator leads to a partial preference of making the same over opposite decisions. In this case we can introduce a parameter |3 c [0, 1] and the utility array is given by:

[0062] Assuming again an independent Bernoulli distribution for the input distribution with parameter p, we can numerically evaluate the quantum advantage. The result isshown in FIGS. 3A, 3B and 3C for a grid of values for p, / 3, as well as different crosssections.

[0063] FIG. 3A shows the quantum advantage for the hedging problem as a function of p and 3, where each ranges from 0 to 1 in increments of 0.1. Shading is in log scale. White squares indicate that there was no quantum advantage found up to floating point error. FIG. 3B shows the quantum advantage as a function of 3 for p = 0.5. And FIG.3C shows the quantum advantage as a function of p for / 3 = 0.4.

[0064] These and other numerical results are obtained via a general-purpose numerical optimizer applicable to any TC problem outlined in Appendix B. We see that when 3 = 0, we obtain the previous results where there is a quantum advantage for:

[0065] Interestingly, as / 3 increases to 0.5, the range of p for which there is a quantum advantage increases, although the value of the quantum advantage decreases. At (3 = 0.5 there is no quantum advantage. For 3 > 0.5, we observe a mirror image by taking / 3 — > 1 - (3. This is because the corresponding utility array can be obtained by appropriately relabeling the observations and decisions, which will lead to the same gap as per Lemma 7 in Appendix A. Hence, we see a family of real-world scenarios where quantum entanglement can provide an indubitable advantage. For convenience of reference, we will refer to this TC problem with the utility array and with an independent Bernoulli distributed input with parameter p as the hedging problem.

[0066] Section 4: Physical Implementation

[0067] In this section we explore the concrete details of physically implementing a quantum strategy for a latency TC problem. We will in particular take the hedging problem as a case study. Our results suggest that there is a tantalizing possibility for quantum entanglement to bring practical gains to real-life TC problems such as those encountered in HFT with current or near-future quantum technologies. Moreover, we have a concrete and realistic roadmap to realize said near-future quantum technologies, such as vacuum beam guides.

[0068] In general, we will classify possible physical implementations into two types: direct photonic connection (Type I) and quantum memory (Type II).

[0069] Section 4.1: Direct Photonic Link

[0070] With reference to FIG. 4, in a first embodiment a system for tacit coordination of decisions 10 comprises:a device 20 for generating entangled photon pairs 30 (an entangled photon pair 30 comprises a first photon 30a and a second photon 30b that are entangled with each other) having a measurable state;a first measurement device 40;a second measurement device 50;a first quantum channel 60 photonically connecting the device 20 for generating the entangled photon pair and the first measurement device 40; and a second quantum channel 70 photonically connecting the device 20 for generating the entangled photon pair and the second measurement device 50.

[0071] The first measurement device 40 is configured to measure the measurable state of first photons 30a of the entangled photon pairs 30 generated by the device 20 for generating the entangled photon pair 30 that are successfully received by the first measurement device 40 based in part on a measurement setting determined using first local information 80 and thereby obtain a first measurement result. Similarly, the second measurement device 50 is configured to measure the measurable state of seconds photon 30b of the entangled photon pairs 30 generated by the device 20 for generating the entangled photo pair 30 that are successfully received by the second measurement device 50 based in part on a measurement setting determined using second local information 90 and thereby obtain a second measurement result. To make tacitly coordinated decisions that achieve a quantum advantage, the first measurement device 40 is configured to make a first local decision 100 using a predetermined quantum strategy based on the first measurement result when the first photons 30a of the entangled photon pairs 30 generated by the device 20 for generating the entangled photon pairs are successfully received by the first measurement device 40 and using a predetermined deterministic strategy when the first photons 30a of the entangled photonpairs 30 generated by the device 20 for generating the entangled photon pairs 30 are not successfully received by the first measurement device 40 due to photon loss in the first quantum channel 60, and the second measurement device 50 is configured to make a second local decision 110 using the predetermined quantum strategy based on the second measurement result when the second photons 30b of the entangled photon pairs 30 generated by the device 20 for generating the entangled photon pairs 30 are successfully received by the second measurement device 50 and using the predetermined deterministic strategy when the second photons 30b of the entangled photon pairs 30 generated by the device 20 for generating the entangled photon pairs 30 are not successfully received by the second measurement device 50 due to photon loss in the second quantum channel 70. The predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account. The quantum advantage is obtained, at least in part, because tacitly coordinated local decisions 100, 110 are made notwithstanding that the distance L between the first and second measurement devices 40, 50 is greater than it takes for light to travel between them (in other words, L is greater than the speed of light c multiplied by time t).

[0072] This can be done at via polarization entanglement (e.g., a type of physical encoding), where there is a continuous wave laser of frequency f being directed at a nonlinear crystal. The nonlinear crystal converts a photon of frequency finto a pair of entangled photons each with frequency f / 2. This is called spontaneous parametric down-conversion (SPDC). This process occurs with some efficiency (only a small fraction of frequency f photons are successfully down-converted). Then, the pair of photons is sent out, one to the first measurement device and one to the second measurement device. The generation rate could be increased for instance by shortening the nonlinear crystal, increasing laser power and / or increasing the number of entangled photon sources.

[0073] In one embodiment, the device 20 for generating the series of entangled photon pairs 30 is configured to perform spatial mode encoding with a fixed linear optical module. In another embodiment, the device 20 for generating the series of entangled photon pairs 30 is configured to perform encoding of cluster states via electrons emitting photons. In another embodiment, the device 20 for generating theseries of entangled photon pairs 30 is configured to perform bosonic quantum encoding. Preferably, the device for generating the entangled photon pairs is configured to perform bosonic quantum encoding that is robust against photon loss errors. More preferably, the system is configured to use bosonic quantum error correction to suppress photon loss errors of up to 50% loss to deterministically generate entanglement. Preferably, quantum error correction is performed before the measurement to suppress loss error.

[0074] In yet another embodiment, the device 20 for generating the entangled photon pair 30 comprises an optically pumped nonlinear crystal. In a preferred embodiment of the invention, the device 20 for generating the entangled photon pair 30 is configured to perform time-bin encoding with fast linear optics.

[0075] As noted above, the entangled photon pairs 30 travel from the device 20 for generating the entangled photon pair 30 to the first and second measurement devices 40, 50 via first and second quantum channels 60, 70. The quantum channels can be, for example, optical glass fiber, quantum repeaters, vacuum beam guides, optical wave guides, air or any other low-loss medium). In one embodiment, the first and / or second quantum channels 60, 70 are vacuum beam guides. In another embodiment, the first and / or second quantum channels 60, 70 are optical fibers. In yet another embodiment, the first and / or second quantum channels 60, 70 are free space (i.e., a vacuum), which includes confined vacuum chambers as well as unconfined natural vacuums and / or near vacuums such as the Earth's exosphere and outer space. In yet another embodiment, the first and / or second quantum channels 60, 70 are optical wave guides.

[0076] FIG. 5 shows greater detail of the subject matter appearing within the dashed box 120 in FIG. 4 (The second measurement device 50 functions in the same manner as the first measurement device 40, but using the second photon 30b of the entangled photon pair 30 and second local information 90). Consider, for example, that the measurable state of the entangled photon pair 30 is that their polarization degrees of freedom are entangled. In this situation, the first measurement device 40 comprises a waveplate 130 and a polarizing beam splitter 140 that routes photons received via the first quantum channel 60 with different polarization to different respective photon detectors 150a, 150b or 150c. It will be appreciated that the number of photondetectors in the first measurement device 40 could be less than or more than the three photon detectors shown in FIG. 2. The waveplate 130 is tunable based on the first local information 80. For example, in the HFT case, if an indicator of negative correlation is observed, the waveplate axis can be set to 30 degrees from vertical, and if an indicator of negative correlation is not observed, the waveplate axis can be set to 60 degrees from vertical. It will be appreciated that the specified angles are exemplary. The tacitly coordinated decision 100 made by the first measurement device 40 is based upon which of the different respective photon detectors 150a, 150b or 150c, measured the measurable state of an entangled photon 30a of the entangled photon pair 30.

[0077] By way of example, presume that the local information a single bit (which may be part of a bit stream). The single bit must have one of two different values, 0 or 1. In this example, the two values determine whether the measurement device should be set such that an entangled photon having a preselected polarization reaches one or the other of the photon detectors 150a, 150b (150c is not used in this simplified example). In this simplified example, the measurement result is whether one of photon detectors behind the polarizer measures the measurable state of an entangled photon or not. If photon detector 150a measures the measurable state of an entangled photon, a first local decision is made (e.g., a processor makes a computation on the local information using a predetermined computation). If photon detector 150b measures the measurable state of an entangled photon, a first local decision is also made. If both photon detectors fail to measure the measurable state of an entangled photon, the second local decision is made. It will be appreciated that the local decision that is made using the local information will vary depending upon the number of detectors being used. This whole procedure, from incoming data reception to computation decision, happens on the timescale of nanoseconds.

[0078] The measurable state of the entangled photon pair could be that their spatial mode encodings are entangled. In such a situation, the first measurement device 40 would be configured to convert spatial mode encoding to polarization, and the first measurement device 40 would comprise a waveplate 130 and a polarizing beam splitter 140 that routes photons with different polarization to different respective photon detectors 150a, 150b or 150c

[0079] The system is suitable for use where the first measurement device and the second measurement device are spaced apart from each other. For example, the first measurement device and the second measurement device can be configured to provide tacitly coordinated local decisions to a first server and a second server, respectively, that are spaced apart from each other such that their decisions have to be made after their observations within a time less than their communication latency. In one implementation, the first measurement device is located in a colocated server of a first stock exchange and the first local information comprises a real-time market data feed from the first stock exchange, and the second measurement device is located in a colocated server of a second stock exchange and the second local information comprises a real-time market data feed from the second stock exchange. The system can be used to facilitate tacitly coordinated trade decisions executed at the respective first and / or second stock exchanges, which is particularly useful in high-frequency trading (HFT). Throughout the instant specification and in the appended claims, HFT refers to computer-implemented trading programs that transact orders in fractions of a second based on market conditions. The term is not limited to arbitrage trades, but includes arbitrage trades. HFT trading often involves the trading of stocks, commodities, and currencies, but can be accomplished with any asset. In some implementations, the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first server and a second server, respectively, that are spaced apart from each other such that their decisions have to be made after their observations within a time less than their communication latency.

[0080] Another implementation of the system shown in FIG. 4 is schematically illustrated in FIG. 6. In this implementation, two processors A and B are shown. The processors may be located in the same large data center or in two neighboring data centers. The processors in this example are about a hundred meters apart and each is receiving half of a stream of GB / s data coming from the Internet. A preprocessor C is configured to split the stream of GB / s data because it is too large for one single processor to handle. The processors (A and B) are configured to make computation decisions at GHz frequencies, while light takes around 1 ps to traverse between the twoprocessors, which means it is impossible for them to communicate with each other at the decision frequencies to coordinate what best pair of computations to run given both input streams. This then defines a TC problem for which systems according to the invention can be applied. Due to loss in the quantum channels, only a fraction of the photons makes it to their destination. If the loss is sufficiently low and the rate of entangled photon generation is sufficiently high, on average there is a photon pair arriving at the measurement devices associated with processors A and B, respectively, every nanosecond. This is the timescale desirable (but not strictly necessary to obtain a quantum advantage) to perform computations with GHz clock speeds. Thus, the systems according to the invention can further comprise a preprocessor that divides a data stream into a first fraction that comprises the first local information and a second fraction that comprises the second local information. In such systems, the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first processor and a second processor, respectively, at a time scale less than their communication latency.

[0081] Systems can also be utilized in computer architecture applications. For example, the first processor and the second processor can have a shared memory and the system minimizes address collisions. To demonstrate some of the concepts and logic that could be involved in a TC problem in a realistic computer architecture scenario that admits a quantum advantage, please consider the following “pseudo-example” (which Applicant admits is inconsistent with how modern computer processor technologies work). Applicant provides this example because it is possible that computer processor technologies could change in the future in a way that makes this example more relevant.

[0082] In this example, there are two CPU’s synchronously making read accesses to memory. We assume that this is happening at such short timescales such that the CPU’s do not have time to communicate with each other. We make two somewhat ad hoc assumptions:1. There are two copies of the memory. Either memory can be accessed by the CPU’s; and2. It is preferable that the CPU’s read from the same memory.

[0083] These assumptions are difficult to justify given the current technology. But continuing, assume CPU A either accesses address 1 or 2 in memory, while CPU B either accesses address 2 or 3 in memory. This setup is shown in FIG. 7. Now, when the two CPU’s want to query different addresses, by Assumption 2 they should read from the same memory. However, if they want to query the same address, they should read from different memories, as reading from the same memory would lead to an address collision. Assuming this is a simple binary game (utility matrix entries are 0 or 1), this setup therefore gives rise to the utility matrix shown in FIG. 8, where the addresses to be queried by the respective CPU’s label the row indices while the memories that are read label the column indices. Up to relabeling, this is exactly the CHSH game.

[0084] Various factors may be configured to enable this protocol including, for example, type of entangled state used, the pairing of waveplate angles to different input data, and the relationship between expected utility for a pair of computation decisions given both input data streams. Under appropriate configurations, the protocol can provably outperform anything that can be done classically (each processor makes its own independent decisions). This can be checked by a numerical optimizer that explicitly parameterizes possible quantum projective measurements with a set of quantum dimensions for each party. Optimization may be performed over both continuous and discrete variables. The utility for the TC problem using this quantum telepathy scheme with experimental imperfections such as loss is then optimized.Various choices of optimizers (e.g., gradient descent, evolutionary algorithm, etc.) may be employed depending on the utility array. In this embodiment, the photon loss from the generator to the measurement devices associated with processors A and B, respectively, may place limits on the performance of quantum protocols. The main challenge is that the measurement devices associated with processors A and B, respectively, are continuously receiving photons, but do not know a priori if the photon entangled with the photon received was successfully received at the other side. Hence, in general, a fraction of these computation decision pairs will not be correlated in the way desired. This can be included in the numerical optimization.

[0085] For loss beyond a certain threshold there is no quantum protocol that achieves a utility higher than that of any classical protocol. This upper bound may be manageable for distances up to a few kilometers for optical fiber (the direct photonic link protocol may still work if the optical fiber is replaced with something with a lot lower loss, such as a vacuum beam guide), but beyond that a different solution may be desirable.

[0086] As previously noted above, we currently live in a world where the speed of light delay between different parties can be appreciable compared to the timescales in which decisions need to be made. Classical processors have GHz clock speeds, while light in vacuum can only travel 30 cm in 1 ns. Hence, TC problems can easily arise in real-world scenarios where communication is not possible due to latency. In general, a TC problem has the following criteria:1. Multiple parties are involved that each make local observations and decisions.2. There is a global utility associated with a set of decisions given a set of observations.3. The parties do not have enough time to communicate their observations with each other before having to make a decision (or they cannot communicate due to other reasons such as the signal being down, a natural disaster, etc.).

[0087] The last criterion could be due to fundamental physics constraints such as speed of light (in vacuum) delay or other factors such as having only an optical fiber connection through which light has to travel farther than the displacement between the parties and travels more slowly than in vacuum. As further described below, quantum entanglement can be utilized to provide an advantage in many scenarios, one of which is high-frequency trading (HFT).

[0088] Two key specifications allow for successful implementation of a quantum strategy for a latency TC problem:1. Entangled photon generation rate2. Photon loss.

[0089] A TC problem arises due to the extremely short time window in which decisions have to be made. Furthermore, in many applications this TC problem repeatsitself with high frequency. For example, in the HFT scenarios previously discussed above, a market maker such as Alpha would be solving the same TC problem over and over again because their method of taking a favorable position or getting returns is always the same. The hedging problem is one such example. For cases where the utility matrix changes with time, the parties and entangled photon source could synchronize changing the quantum strategy over time. Nevertheless, this leads to the first requirement for the physical setup: the entangled photons are generated at a sufficiently high frequency so that a quantum strategy can be executed within the time window. In HFT, these time windows are on the order of microseconds, which means entangled photons need to be generated at MHz rates. This is possible with current technologies using a continuous-wave laser and a nonlinear crystal to produce photon pairs maximally entangled in polarization via parametric down-conversion. These pairs can be produced at MHz rates. It is possible to accommodate even shorter time windows by using pumped pulses, thereby achieving generation rates up to 50 GHz. One could also increase the number of entangled photon sources. However, note that it is not strictly necessary that the entanglement generation rate be sufficiently high to attain a quantum advantage overall. The parties could simply execute an optimal deterministic strategy during times when entanglement is unavailable, and execute an optimal quantum strategy when it is available.

[0090] The second specification is photon loss. This is a key difficulty for a direct photonic link implementation that could even preclude a quantum advantage. Because of photon loss, not only are photons sometimes unavailable, each party does not know whether the other party actually received their entangled photon. One could imagine that if they both knew that one of the photons was lost, they could revert to the best deterministic strategy and thereby effectively achieve a nontrivial convex combination of the classical and quantum values. This would retain a quantum advantage. However, because they each do not know whether a photon was lost for the other party, they only have probabilistic control over what strategy they implement. The Appendices explains how exactly loss affects the resulting behavior of a quantum strategy. In general, the resulting behavior will be a convex combination of behaviors where the weight is the probability of entanglement loss and the parties that experience loss fall back to adeterministic strategy. As the probability of loss increases, the quantum behavior eventually becomes a classical behavior, thereby leading to a loss of a quantum advantage.

[0091] Given a TC problem, assume each party successfully obtains their entangled particle with probability q, which we call the efficiency. We can then define a threshold efficiency q* as the minimal value such that for q > q*, we can attain a quantum advantage. If the TC problem had no quantum advantage even without loss, we define q*:= 1. Consider for example the hedging problem. We can use the numerical optimizer outlined in Appendix B to compute q* for different possible values of p and (3. This is shown in FIGS. 9A, 9B and 9C. In particular, FIG. 9A shows 1 - q* values computed for the hedging problem with p and p taking values between 0 and 1 with increments of 0.1. Shading is in log scale. When there was no quantum advantage to begin with, by definition 1 - q* = 0 and we draw a white square. FIG. 9B shows q* as a function of 3 when p = 0.5. The lower bound of2 / 3 is plotted as a dashed line. And, FIG. 9C shows q* as a function of p when 3 = 0.4. At a qualitative level, we see the same pattern as in FIGS. 3A-3C. This is intuitive as lowering the efficiency will continuously lower the achievable expected utility for quantum strategies with loss until it reaches the classical value.

[0092] It is known that for TC problems with two parties where each party has two possible observations, the threshold efficiency is at least2 / 3. This is also apparent in the plots. If we use optical fiber to distribute the photons, this efficiency is already difficult to achieve for the distances involved for HFT. In general, q exponentially decreases with the length of the channel / :η = e-αl.where a is the attenuation rate of the physical medium used for the channel.

[0093] Now, the lowest attenuation rate achievable by optical fiber for the optimal wavelength of 1550 nm is 0.17 dB / km. Hence, to achieve an efficiency of at least2 / 3, the maximum length I allowed is around 10.4 km. Note that this is the distance from thesource to each party is therefore half the distance between the two servers. The efficiency computed should also include detector efficiency, inter-component coupling efficiency, and other possible sources of photon loss. This places an even stronger bound on length. We see q* values considerably higher than2A in the plot, which shows that using optical fiber in a direct photonic link implementation is insufficient for the hedging problem when the two servers are separated by larger distances, such as in the case of NYSE to NASDAQ. There are other physical medium to consider, however. Vacuum beam guides could be used to distribute entangled photons with which an attenuation rate as low as 5 x 1O“5dB / km can be achieved according to numerical simulations. In this case, an efficiency of2 / 3 is achievable for distances less than about 35,000 km, which is about the circumference of the Earth. Thus this would allow for quantum strategies to be executed at continental distance scales, for which speed of light delays can be significant. For example, the geodesic distance between the NYSE data center and the Hong Kong Exchanges and Clearing Limited (HKEX) data center is about 12.9 x 103km, which is a speed of light delay of about 43.1 ms. This is very long compared to the time scale of HFT. For the case of NYSE and NASDAQ, using vacuum beam guides we can achieve 1 - q of about 3.24 x 10-4, which is sufficient for the majority of the points. Note that optical fiber can still be useful for TC problems with shorter distances, or with a higher number of parties or possible observations.

[0094] For example, finer details of strategies for the hedging problem, setting p = 0.3, 3 = 0.3 for concreteness. In this case the threshold efficiency is computed to be q* ~ 0.941. So that we attain a noticeable quantum advantage, we set q = 0.95. We run the numerical optimizer and find that the classical value is c* = 0.79, where an optimal deterministic strategy is for the NYSE server to always bid first and the NASDAQ server to ask first when it sees no indicator and to bid first when it sees an indicator. The highest expected utility for a quantum strategy with efficiency q = 0.95 issf; •••.•• 05) 0,792.

[0095] We prove in Appendix A3 herein that when there are only two possible observations and decisions for each party, qubit systems are sufficient for achieving the quantum value, even in the presence of loss. Furthermore, we can parameterize the measurement basis with only one parameter.

[0096] Define the states:sin 109)

[0097] One optimal quantum strategy uses the following measurement operators. When neither server sees an indicator, both servers measure in the computational basis (Since we only care about the largest eigenvalue of the Bell operator, there is a local unitary degree of freedom and therefore we can without loss of generality always assume the measurement for the first observation is in the computational basis){|0><0|, |1)(1|}.

[0098] When either server sees an indicator, it instead uses the measurement| ""0.590) L k'"'" ("" 0- SOO)) (w1( • • O. Wn i |,

[0099] where the angle is in radians and three significant figures are kept. The shared entangled state isi) ™ 0.0401 KM) "" 0.9024)1) "" 0.423(10 ■■■• 0.04011)1},keeping three significant figures for the coefficients. Note that since the measurement operators have all real elements, the shared entangled state also has all real elements. To realize this quantum state physically, we perform a Schmidt decomposition:

[0100] wherehp) (L990510) ••}• 0,0301 h.khM t™ O. OSOIJO). G.ikWll)

[0101] andkp) '4T13W31I), IMJ) I-~ —01 WS 10} 4' 0.030111).

[0102] The state in equation (4.1 ) can for example be realized by letting

[0103] be the vertical-horizontal polarization basis and using the technique of Marissa Giustina, Alexandra Meeh, Sven Ramelow, Bernhard Wittmann, Johannes Kofler, Jbrn Beyer, Adriana Lita, Brice Calkins, Thomas Gerrits, Sae Woo Nam, et al. Bell violation using entangled photons without the fair-sampling assumption. Nature, 497(7448):227-230, 2013. The single-qubit rotations needed before the measurement to switch back to the computational basis can be realized using linear optics. Lastly, the optimal fallback deterministic strategy is for the NYSE server to always ask first and the NASDAQ server to always bid first.

[0104] The present invention also includes methods for making tacitly coordinated decisions given first and second local information, respectively. With reference to FIG.10, in one embodiment, the method comprises:generating entangled photon pairs having a measurable state;transmitting, via first and second quantum channels, respectively, first photons of the entangled photon pairs to a first measurement device and second photons of the entangled photon pairs to a second measurement device; measuring, using the first measurement device, the measurable state of the first photons of the entangled photon pairs that are successfully received by the first measurement device based in part on a measurement setting determined using first local information to thereby obtain a first measurement result;measuring, using the second measurement device, the measurable state of the second photons of the entangled photon pairs that are successfully received by the second measurement device based in part on a measurement setting determined using second local information to thereby obtain a second measurement result;to achieve the quantum advantage, making a first local decision via the first measurement device using a predetermined quantum strategy based on the first measurement result when the first photons of the entangled photon pairs are successfully received by the first measurement device and using a predetermined deterministic strategy when the first photons of the entangled photon pairs are not successfully received by the first measurement device due to photon loss in the first quantum channel, and making a second local decision via the second measurement device using the predetermined quantum strategy based on the second measurement result when the second photons of the entangled photon pairs are successfully received by the second measurement device and using the predetermined deterministic strategy when the second photons of the entangled photon pairs are not successfully received by the second measurement device due to photon loss in the first quantum channel, wherein the predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account.

[0105] In one implementation of the method, the first local information comprises a real-time market data feed from a first stock exchange, and the second local informationcomprises a real-time market data feed from a second stock exchange. In accordance with the method, tacitly coordinated decisions are made faster than the communication latency between the first stock exchange and the second stock exchange. Preferably, the real-time market data feed from the first stock exchange and the real-time market data feed from the second stock exchange include data that indicates a statistical relationship between a plurality of financial instruments. The tacitly coordinated decisions can be trading decisions, including but not limited to, trading decisions are used for hedging.

[0106] In another example of the method, the tacitly coordinated decisions are computations made by a plurality of servers performing distributed computing. In another example, the tacitly coordinated decisions can be, for example, instructions executed by separate processors. The separate processors can, for example, read from a shared memory and the tacitly coordinated decisions minimize address collisions and / or the first and second local information comprise addresses in the shared memory that the separate processors are to query.

[0107] Section 4.2: Using Quantum Memory

[0108] The main shortcoming of direct photonic link (Embodiment A) implementations is that the magnitude of photon loss in optical fiber, the predominant physical medium for photonic communication in industrial applications, is too high for many TC problems. Embodiment B solves this problem by having the parties each have a quantum memory, which could be a simple (few qubits) quantum computer with a long coherence time. The memories are entangled via photons in a heralded entanglement scheme. Because the entanglement is heralded, we can avoid the problem in Embodiment A implementations where the parties do not know if entanglement was successfully distributed. Furthermore, by using quantum memories we can realize more complicated and higher dimensional entangled states that may be difficult to realize with photons.

[0109] The main specs we need to consider for Embodiment B implementations are 1. Effective entanglement generation rate: re2. Fidelity of quantum operations: F.

[0110] The effective entanglement generation rate is determined by the following parameters:Entanglement attempt time: taHeralded entanglement success probability: psQuantum memories multiplicity: M

[0111] Here, tais the time needed to generate entanglement for each quantum memory. This includes the time for the physical operation to create memory-photon entanglement, the time for the photons to travel to the intermediate node, transduction time, and the time for the heralded entanglement information to be received. Usually tais dominated by the traversal times. Next, psincludes the probability of successfully creating memory-photon entanglement, the probability of the photons reaching the intermediate node, and the probability of correctly projecting the photons into the desired state. Lastly, M is the number of quantum memories available to each party. This increases the effective entanglement generation rate by a multiplicative factor. In summary,

[0112] We would like a Type II implementation to have an effective generation rate sufficiently high for the TC problem’s demand for entanglement (although again like Type I implementations this isn’t necessary to get an overall quantum advantage).

[0113] We compute the effective entanglement generation rate for HFT between NYSE and NASDAQas a function of M. Assuming tais dominated by photon traversal time, we split it into two terms. The entangled photon traversal we will assume is through fiber with velocity Vf, while to minimize time for heralding we can use free space transmission with velocity vs. Then,where we takeand

[0114] Next, we assume psis dominated by photon loss and the probability of successful photon state projection which is usually1 / 2. Thus, using the optical fiber attenuation rate of 0.17 dB / km,’

[0115] We therefore have• 10 Hz.

[0116] For Type II implementations, fidelity is also an important specification.Entanglement generation involves noisy memory-photon entanglement, imperfect photon detection, as well as decoherence of the memories during photon traversal. Measurement of the memories themselves can also be noisy. We will evaluate how much the quantum advantage computed for the hedging problem previously plotted above is robust to noise. For simplicity, we will assume depolarizing noise on the entangled state, but in realistic scenarios, a detailed physics simulation should be conducted. We define the robustness v* as how much depolarizing noise can be tolerated before the quantum advantage disappears.

[0117] Now, the effect of depolarizing noise

[0118] where TT is the maximally mixed state and v is the magnitude of the noise, on the behavior of a quantum strategy for the hedging problem is the following:H? 44 •: I C S Z- J- -- U >

[0119] where we assume the quantum strategy achieves the quantum value and uses only qubits. Using higher dimensional systems could affect the robustness, but at the cost of also incurring larger error rates for the physical implementation. For aderivation of these results, see Appendix A4. Using eq. (4.2) and the utility array eq. (3.1), we can directly compute the effect of depolarizing noise on the expected utility:

[0120] Conveniently, this transformation does not depend on p or 3. We observe the second term in eq. (4.2) corresponds to a constant classical behavior where each party independently chooses one of the possible decisions A, B with probability1 / 2 regardless of the observation, so the utility of1 / 2 achieved by such a behavior as computed in eq. (4.3) is always at most the quantum value. Thus, we can simply compute the robustness as:

[0121] FIGS. 11A-11C show the robustness for the hedging problem as a function of p and 3. Specifically, FIG. 11A shows robustness values computed for the hedging problem with p and 3 taking values between 0 and 1 with increments of 0.1. Shading is in log scale. When robustness is 0, we draw a white square. FIG. 11 B shows robustness as a function of 3 when p = 0.5. And FIG. 11C shows robustness as a function of p when 3 = 0.4.

[0122] In the range of p and 3 considered, other than some extreme values, we observe a robustness to depolarizing noise of about 10’3to 10’1. We even see robustness values of up to 0.293. That is, optimal quantum strategies using qubits can tolerate depolarizing noise up to (not including) that magnitude and a quantum advantage would remain. For fixedwe also observe some regions with constant robustness, which implies in those regions the quantum value q* is a fixed affine function of the classical value c*. In general, noise robustness can be interpreted as some form of normalized maximal Bell inequality violation. FIGS. 12A-12C plot the quantum advantage for different values of p and 3 under various noise levels.Specifically, FIG. 12A shows the quantum advantage for the hedging problem as a function of p and 3, where each ranges from 0 to 1 in increments of 0.1, for variouslevels of depolarizing noise. Shading is in log scale. White squares indicate that there was no quantum advantage found up to floating point error. The range of p and 3 for which there is a quantum advantage shrinks as the noise level increases as shown in FIGS. 12B and 12C

[0123] With reference to FIG. 13, in a second embodiment a system for tacit coordination of decisions 210 comprises:a first measurement device 220;a second measurement device 230;a first of a pair of quantum memory devices 240 coupled to the first measurement device 220;a second of the pair quantum memory devices 250 coupled to the second measurement device 230;a heralded entanglement measurement device 260;a first quantum channel 270 photonically connected between the heralded entanglement measurement device 260 and the first quantum memory device 240; anda second quantum channel 280 photonically connected between the heralded entanglement measurement device 260 and the second quantum memory device 250.

[0124] The first and the second of the pair of quantum memory devices 240, 250 are each configured to repeatedly emit photons 290a, 290b, respectively, toward the heralded entanglement device 260 via the first and second quantum channels 270, 280. It is expected that some of the photons 290a, 290b will be lost or not received by the heralded entanglement device for various reasons disclosed herein. The heralded entanglement device 260 is configured to detect photon loss errors. The heralded entanglement measurement device 260 is configured to transmit heralding signals 320 to the first and second quantum memory devices 240, 250, respectively, indicating when the first and second quantum memory devices 240, 250 have been successfully entangled. The first measurement device 220 is configured to perform a measurement of the first quantum memory device 240 based in part on a measurement setting determined using first local information 300, and the second measurement device 230 isconfigured to perform a measurement on the second quantum memory device 250 based in part of a measurement setting determined using second local information 310.The first and second measurement devices 220, 230 are configured to make the tacitly coordinated decisions 330, 340 based on the measurements made by first and second measurement devices 220, 230, thereby obtaining a quantum advantage. In a preferred implementation, the system 210 further comprises a first quantum transducer 350 coupled to the first of the pair quantum memory devices 240 and a second quantum transducer 360 coupled to the second of the pair quantum memory devices 250. The first and second quantum transducers 350, 360 are configured to convert the frequencies of the photons 290a, 290b emitted by the entangled quantum memory devices 240, 250, while preserving entanglement, for transmission through the first and second quantum channels 270, 280, respectively.

[0125] The first and second measurement devices 220, 230 are preferably configured to make the tacitly coordinated decisions using a predetermined optimal quantum strategy when they determine that there are sufficient entangled memory devices to do so, and are configured to make the tacitly coordinated decisions using a predetermined optimal deterministic strategy when they determine that there are insufficient entangled memory devices to do so.

[0126] To suppress depolarizing errors, the system is preferably configured to use an entanglement distillation procedure, which uses multiple Bell pairs and local operation and classical communication to distill Bell pairs with higher fidelity. This procedure is described, for example, in the following papers, which are hereby incorporated by reference for their teachings: (1 ) " Concentrating partial entanglement by local operations," C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996); (2) " Optimized Entanglement Purification," S. Krastanov, V. V. Albert, and L. Jiang, Quantum 3, 123 (2019); and (3) " Quantum repeaters: From quantum networks to the quantum internet," K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, Reviews of Modern Physics 95, 045006 (2023). The distilled Bell pairs can be used for teleported gates to obtain logical Bell pairs with quantum error correction protection, for example, as disclosed in " Quantum Repeaterwith Encoding," L. Jiang, J. M. Taylor, K. Nemoto, W. J. Munro, R. Van Meter, and M. D. Lukin, Phys. Rev. A 79, 032325 (2009), which is hereby incorporated by reference.

[0127] The quantum channels 270, 280 used in this embodiment can have the same structure as those described in the first embodiment (Embodiment A).

[0128] The first and second measurement devices 220, 230 preferably each comprise a camera for detecting the state dependent fluorescence.

[0129] The system 210 preferably comprises first and second lasers configured to drive the first and the second of the pair of quantum memory devices 240, 250, respectively, to emit the photons 290a, 290b that the heralded entanglement device 260 receives via the first and second quantum channels 270, 280. The system 210 also preferably further comprises a first and second microwave drives configured to perform quantum gate operations on the first and second of the pair of quantum memory devices 240, 250 based on local information 300, 310 when the heralding signal 320 indicates that the quantum memory devices 240, 250 have been successfully entangled. And, the system further preferably comprises first and second measurement lasers directed at the first and second of the pair of quantum memory devices 240, 250 that generate state dependent fluorescence which the first and second measurement devices 220, 230 are configured to detect to make the tacitly coordinated decisions 330, 340.

[0130] In one implementation, the pair of quantum memory devices 240, 250 comprise at least one of:atoms;ions;superconducting qubits;nitrogen vacancy centers;nuclear / electron spins; andmicrowave cavities.

[0131] In one implementation, the first measurement device 220 is located in a colocated server of a first stock exchange, and the first local information 300 comprises a real-time market data feed from the first stock exchange. In this implementation, the second measurement device 230 can be located in a colocated server of a secondstock exchange, and the second local information 310 can comprise a real-time market data feed from the second stock exchange. Thus, the tacitly coordinated decisions 330, 340 can be trade decisions executed by the servers at the respective first and / or second stock exchanges.

[0132] In one implementation of Embodiment B, a system includes two servers (A and B, respectively) located at a distance from each other. In an example use case, the servers may be engaged in HFT; one in a NYSE data center and one in a NASDAQ data center as shown in FIG. 2. The data centers of the two exchanges are about 56.3 km apart. Light takes about 188 ps to traverse between the two servers, while HFT is conducted on the time scale of ps, so the servers cannot communicate at these time scales. The described system uses quantum entanglement to get an advantage for the tacit coordination (TC) problem that is defined by the utility, or expected return, of the pair of trades made by the two servers given their respective local inputs such as a price of a stock at NYSE or at NASDAQ going up or down.

[0133] A practical way to establish entanglement at such distances is to use n pairs of neutral atom quantum memories (in this example, physical quantum memory), one of each pair at each server. That is, server A has n atoms, each of which is paired with one of n atoms at server B. Each atom at each server is photonically connected via a network of optical switches to an optical fiber cable (or other low-loss quantum channel) from each server to an intermediate node S that is between A and B. As in the prior embodiment, measurement devices A and B, respectively, are in close physical proximity to servers A and B, respectively.

[0134] Now, every pair of atoms is continuously trying to generate entanglement with each other. Each atom does this by being excited by a laser, thereby emitting a photon that is entangled with the atom (in this example, this determines a physical encoding of entanglement in photons). The emitted photon is usually in the optical wavelength, so it goes through a quantum transducer which converts its frequency to the telecom regime in a way that preserves the entanglement with the atom. The converted photon then goes through the optical fiber (with minimal loss due to the converted wavelength). If synchronized correctly, at node S there should arrive simultaneously a pair of photons, each photon entangled with its corresponding atom. An appropriate photon detection(which happens with some probability) at the two output modes of the beamsplitter causes the pair of atoms with which the photons were entangled to become themselves entangled with each other (the photons are now gone). A signal is sent from S to A and B notifying them of a successful or unsuccessful entanglement generation event of the corresponding pair of atoms.

[0135] In HFT, although the “reaction time” (the time between local observation and local trade decision) is on the order of microseconds or less, the “event time” (time between events that trigger a trade decision) is on the order of milliseconds or even longer. Hence, even for n=1, with the 28km distance from A to S and B to S and the corresponding large photon loss, the entanglement generation rate is still fast enough to yield an entangled pair of atoms every few milliseconds. This is sufficient for HFT. Such single-memory technologies are already available. This can also depend on the rate at which each pair of atoms can attempt to become entangled, and also how long entanglement can last once generated.

[0136] In operation, the server at A and B each can observe price changes at NYSE and NASDAQ, respectively. According to what price change they observe, they conduct a corresponding operation on a pair of atoms that were successfully entangled, followed by a measurement. That is, depending on what specific price change they observe, each server will first perform a corresponding quantum gate on the atom, which involves shining it with some sequence of laser pulses to change its quantum state. The measurement involves directly shining a different laser on the atom and then checking for any subsequent fluorescence. The measurement result (fluorescence seen or not seen) then determines a corresponding trade decision, such as buying or selling a certain stock. This whole procedure, from price change observation to trade decision, happens in the time frame of microseconds.

[0137] The protocol may be designed with various configurable parameters such as the type of entangled state used, the pairing of quantum operations to different price change observations, the relationship between expected return for a pair of trades given both price change observations, etc. Under appropriate configurations, this protocol can provably outperform anything that can be done classically (the term "classically" means that each server makes its own independent decisions without coordinating withthe other). This can be checked by a numerical optimizer as mentioned above. Here, the photon loss from optical fiber does not place a fundamental restriction because the system can store entanglement in quantum memories which can be flagged as successfully entangled or not.

[0138] Quantum telepathy is a new application of quantum technologies. In our modem era of electronic computing where timescales are measured in microseconds or even nanoseconds, the speed of light delay is actually appreciable in many scenarios. Coordinating decisions at such timescales between space-like separated parties could benefit from using quantum entanglement. High frequency trading is a notable example.

[0139] Aside from HFT, other possible applications include distributed computing and (classical) computer architecture. While HFT is somewhat specialized, these latter two scenarios are much more common. Distributed computing may involve computers with a spatial separation of around 100 meters, the square root of the area of an average data center in 2024. At this distance, any computational processes faster than 0.1 microseconds would be space-like separated. Current entangled photon sources can achieve generation rates at this timescale and optical fiber can achieve an efficiency of q ~ 0.998 at this distance scale. Hence this is the best regime to make use of current technologies. As for the latter application of computer architecture, distance scales may be on the order of 1 cm or shorter in a CPU (for example an Intel i7 CPU core package size is 50*25 mm). Although it would be difficult to find any processes faster than the speed of light delay at such a distance scale, in general signal delays in a microprocessor cache can be dominated by resistive-capacitive time delays in long on-chip interconnects. Furthermore, modern computer memories can have latencies significantly longer than the speed of light delay: DDR memories can have latencies on the order of 10 ns, while SSD memories can have latencies of 10 to 100 ps. Needing to coordinate processes in different parts of the memory at these timescales may therefore engender a TO problem. However, for such timescales, we may need to use photon sources with very high generation rates such as disclosed by Kentaro Wakui, Yoshiaki Tsujimoto, Mikio Fujiwara, Isao Morohashi, Tadashi Kishimoto, Fumihiro China, Masahiro Yabuno, Shigehito Miki, Hirotaka Terai, Masahide Sasaki, et al. Ultra-high-rate nonclassical light source with 50 ghz-repetition-rate mode-locked pump pulses and multiplexed single-photon detectors. Optics Express, 28(15):22399-22411, 2020, as well as faster electronics for measurement. Now, in the case of computer architecture, for integration purposes it is more practical to use optical waveguides to distribute photons. Compared to optical fiber, waveguides have a much higher attenuation rate of about 0.2 dB / cm, which implies an efficiency of q « 0.955 for a 1 cm distance. Note that the speed of light in a waveguide is around the same as that of optical fiber: 2 / 3 of that in vacuum.

[0140] The present invention also provides methods for making tacitly coordinated decisions given first and second local information. With reference to FIG. 14, in one embodiment, the methods comprise:emitting a pair of photons, said pair of photons including a first photon emitted from a first quantum memory and a second photon emitted from a second quantum memory;measuring the pair of emitted photons to determine whether the first and second quantum memories are entangled;when the first and second quantum memories are determined to be entangled, sending heralding signals to the first and second quantum memories that indicates successful or unsuccessful entanglement;storing the successful entanglement in the first and second quantum memories, respectively;performing quantum operations on the first and second quantum memories given the first and second local information;consuming the stored successful entanglement by measuring the first and second quantum memories; andmaking the tacitly coordinated decisions based on the measurements of the first and second quantum memories to obtain a quantum advantage.

[0141] In one implementation of the method, the first local information comprises a real-time market data feed from a first stock exchange, and the second local information comprises a real-time market data feed from a second stock exchange. In accordance with the method, the tacitly coordinated decisions are made faster than latency incommunication between the first stock exchange and the second stock exchange. The real-time market data feed from the first stock exchange and the real-time market data feed from the second stock exchange preferably include data that indicates a statistical relationship between a plurality of financial instruments. The tacitly coordinated decisions can be trading decisions, including but not limited to, trading decisions are used for hedging.

[0142] In another implementation of the method, the tacitly coordinated decisions are computations made by a plurality of servers performing distributed computing. The tacitly coordinated decisions can be instructions executed by separate processors. The separate processors can read from a shared memory and the tacitly coordinated decisions can minimize read collisions from the memory. In another implementation, the first and second local information comprise addresses in the shared memory that the separate processors are to query.

[0143] For preparing a specific entangled quantum state between multiple parties, we can take multiple approaches. For Embodiment A implementations, one possible approach is for the entanglement source to use quantum computers such as atoms or superconducting circuits to produce optical or microwave entangled photons with a programmable quantum state. For Embodiment B implementations, quantum teleportation is a natural solution. Additional features of the above described system may include any of the following (alone or in any combination):1. Physical implementationa. Physical medium for photon transmission (low-loss quantum channel)i. Vacuum beam guide for 10 km or longer ii. Optical fiber or free space for 1 cm to 10kmiii. Optical waveguide for 1 cm or shorterb. Entangled quantum statei. Photons1. Time-bin encoding with fast linear optics2. Spatial mode encoding with fixed linear opticalmodule3. Polarization encoding for maximally entangled states a. Increasing frequency of photon generation to GHz or higher4. Cluster states via electron emitting photons ii. Quantum memory: atoms, ions, superconducting qubits, NV centers, nuclear spins, microwave cavities1. Heralded entanglement using photons via intermediate stationa. Multi-mode memory (a memory with multiple atoms or others - see above)2. Local operations on memoryMethod for determination of whether quantum advantage exists with loss a. See if loss is above threshold efficiency for N parties and M observationsb. Compute classical value, see if quantum value with loss can exceed itc. Evaluate if frequency of received photons with loss is sufficiently highd. Effect of thermal noise, detector inefficiency, dark counts Specific real-world examples of TC problemsa. High frequency tradingi. CHSH-like game for responding to anti-toxic flow indicators b. Distributed computingc. Computer architecturei. Parallel processors with shared memoryNumerical optimizer (what is the best quantum protocol to use) a. Parameterization of quantum projective measurementsb. Discrete and continuous variable optimization via evolutionary algorithm

[0144] Additional subject matter may include, but is not limited to the following features which may be incorporated alone or in combination with any of the embodiments described herein:1. A method for determining an appropriate physical implementation of a quantum protocol for a tacit coordination problem, comprising a. determining the appropriate physical medium for photon transmission given the distances between parties.b. determining the physical encoding of the entangled state into photons given the parameters of the quantum telepathy scheme including the complexity and dimensionality of the required entangled state, number of observations or measurement schemes, and number of decisions or measurement outcomes. Other factors such as loss and noise, as well as the rate of entanglement may implicate design parameters.c. determining an appropriate quantum memory given the distances between parties and parameters of the quantum telepathy scheme. The quantum memories themselves are entangled via photons at an intermediate node and may involve multiple entangled modes. d. determining whether quantum telepathy can provide an advantage for an TC problem. This method parameters may be based on the distances between parties, the corresponding photon loss, the background noise levels, the rate at which entanglement is needed and is available, and the specific utility array or matrix of the TC problem.2. The method of 1, where the TC problem relates to:a. High frequency tradingb. Distributed computingc. Computer architecture3. The method of 2, where the high frequency trading problem involves using indications of stock price correlation to determine whether to make different trading serves make the same or opposite trading decisions.The method of 2, where the computer architecture problem involves two processors with a shared memory that operate to minimize address collisions.The method of claim 1, where the different physical media for photon transmission comprises at least one ofa. Optical fiberb. Free spacec. Optical waveguidesd. Vacuum beam guidesThe method of 1, where the different possible physical encodings of entangled states into photons comprises at least one ofa. Time-bin encoding with fast linear opticsb. Spatial mode encoding with fixed linear optical modulec. Polarization encoding for maximally entangled statesd. Cluster states via electron emitting photonse. Bosonic quantum encodingThe method of 6, where the entangled photons generation rate is increased by shortening the nonlinear crystal length, increasing laser power, and increasing the number of photon sources.The method of 1, where the quantum memories comprise at least one of a. Atomsb. Ionsc. Superconducting qubitsd. Nitrogen vacancy centerse. Nuclear / electron spinsf. Microwave cavitiesThe method of 1, where the determination of whether a quantum telepathy scheme provides an advantage for a TC problem is realized via a numerical optimizer that explicitly parameterizes possible quantum projective measurements with a set of quantum dimensions for each party.The utility for the TC problem using this quantum telepathy scheme with experimental imperfections such as loss is then optimized.10. The method of 9, where the optimization is done viaa. Brute force searchb. Evolutionary algorithmsc. Gradient descent11. The method of 10, where the optimization involves both discrete and continuous variables.12. The method of 9, where given the number of parties and observations of the TC problem, a lower bound on the loss permitted for a quantum advantage to exist is computed.

[0145] The present invention also provides a method for determining an appropriate physical implementation of a quantum protocol for a tacit coordination problem between at least two parties, the method comprising:determining an appropriate physical medium for photon transmission given a distance between the parties;determining physical encoding of an entangled state into photons given the parameters of the quantum telepathy scheme including the complexity and dimensionality of the required entangled state, number of observations or measurement schemes, and number of decisions or measurement outcomes;determining an appropriate quantum memory given the distances between parties and parameters of the quantum telepathy scheme; and determining whether quantum telepathy can provide an advantage for an TC problem.

[0146] This method parameters may be based on the distances between parties, the corresponding photon loss, the background noise levels, the rate at which entanglement is needed and is available, and the specific utility array or matrix of the TC problem. There are a plurality of quantum strategies that achieve different utilities. The method includes a step of evaluating the plurality of quantum strategies and selecting the optimal quantum strategy that has the highest number of utilities. In one embodiment,the method of determining whether a quantum telepathy scheme provides an advantage for a TC problem is realized via a numerical optimizer that explicitly parameterizes possible quantum projective measurements with a set of quantum dimensions for each party. The utility for the TC problem using this quantum telepathy scheme with experimental imperfections such as loss is then optimized. Optimization can be done via one or more of:a. brute force search;b. evolutionary algorithms; andc. gradient descent.

[0147] Optimization can involve both discrete and continuous variables. And, in accordance with the method, where given the number of parties and observations of the TC problem, a lower bound on the loss permitted for a quantum advantage to exist is computed.

[0148] At least some portions of the described system and corresponding processes may be implemented by one or more classical computing systems that may operate in conjunction with quantum elements. The one or more classical computing systems include at least one processor and a non-transitory computer-readable storage medium storing instructions executable by the at least one processor for carrying out the processes and functions described herein. The computing system may include distributed network-based computing systems in which functions described herein are not necessarily executed on a single physical device. For example, some implementations may utilize cloud processing and storage technologies, virtual machines, or other technologies.Appendix A: General TC Problems: Definitions and Facts

[0149] In this section we present the theory of TC problems. Many of the concepts before Appendix A1 are well known in the literature on Bell inequalities, although we do make some generalizations and slight deviations to capture essential elements of a real-world TC problem and give a convenient framework to prove some results. We make such deviations explicit.

[0150] We first define a general TC problem.

[0151] Definition 1 Let n > 2 be an integer. An n-party TC problem is a tuple:

[0152] where O / , Di are non-empty finite sets (Theoretically, we could consider infinite sets, but this is not feasible in practical experimental settings). po(o) is a probability distribution over O:= Oi x O2x■ • ■ x On, and u is a multidimensional array indexed by observations and decisions:

[0153] We also define D:= Di x o2x ■ ■ ■ x On.

[0154] O(and D;are the sets of observations and decisions for party / . po(o) is the input distribution, a probability distribution over the set of possible observation tuples across all parties. The marginal distributions of observations for each party in general may not be independent. This is an additional element that is present in nonlocal games but not general Bell expressions. Lastly, u is the utility array, where

[0155] is the utility of making a combination of decisions

[0156] given the combination of observations- T O.

[0157] We deviate from known theory by proposing a multidimensional array structure instead of a vector structure for their equivalent notion of a Bell expression. This is more natural since for one we should differentiate the observation indices from the decision indices and each party’s index from each other. We also introduce the terminology that when each party has the same number of observations m and the same number of decisions A, we have an (n, m, ) problem.

[0158] Definition 2. For any? £ 44 44 44 > 4- - P

[0159] We define a behavior as a multidimensional arrayas conditional probabilities p(d\o). The expected utility of a behavior is given by

[0160] The conditional probability distribution p(d\o) describes how parties make decisions given the observations. Again, we deviate from prior art methods by choosing to describe this as a multidimensional array instead of a vector. Now, in a TC problem, the parties are not in communication. Hence, their behavior has to satisfy a nosignaling condition, that is, for all

[0161] in their respective alphabets,

[0162] is the same for all... x-XkA.

[0163] In words, the marginal distribution over decisions of all parties other than party / is independent of the observation of party / . Here, our choice using a array structure for the behavior gives the no-signaling condition a more natural interpretation as a form of multidimensional array symmetry. Note that the TC problem formalism generalizes nonlocal games that usually only have only binary outcomes of winning or losing. The formalism can capture this notion as a special case by lettingif (o, cf) satisfies the winning conditions and equal 0 otherwise. Then, the expected utility of a behavior is the probability of winning the nonlocal game.

[0164] We next define what behaviors are possible for parties with classical and quantum resources.

[0165] Definition s. A deterministic strategy is given by

[0166] In a deterministic strategy, each player has a local function: 0 / — Di which they use to make decision di given observation oz. The corresponding deterministic behavior is given by

[0167] where o:= (01, 02, ■ ■ ■, On), d:= (di, cfe, ■ ■ ■, dn), and 5 is the Kronecker delta function. The expected utility of a deterministic behavior is given bywhere f: 0 — D is given by f((oi, 02, ■ ■ ■, On)):= (fi(oi), £(02), ■ • ■, f (On)). More generally, a classical behavior is defined by convex combinations of deterministic behaviors, which are attained by classical strategies: probabilistic mixtures of deterministic strategies where shared randomness can be used.

[0168] Definition 4. A quantum strategy is a tuple:’ l 7 < J 5 $ S S 7" J:■ S ^'7 j ’where q, > |D / | is a positive integer, Mi:0, |D / j), S / w(q / , |Q|) being the set of all projective measurements on a qi-dimensional Hilbert space consisting of |D / j projectors, andis a

[0169] In a quantum strategy, the players share a global quantum state:

[0170] where each of their shares is of dimension qt. They apply a projective measurement / W / (o) on their share when they make the observation o / . There are |D;j possible measurement outcomes, which correspond to the possible decisions each party can make. We can more conveniently denote the projectors of M(o / ) asindexed by

[0171] The corresponding quantum behavior is then given by

[0172] We can therefore express the expected utility as

[0173] We will call the operatorthe Bell operator of a quantum strategy. For the purposes of computing the quantum value it is sufficient to compute the largest eigenvalue of the Bell operator. Note in general we can also allow density matrices and POVM elements, but such strategies can always be considered to be a quantum strategy according to Definition 4 but in a higher dimensional space via Naimark’s dilation theorem.

[0174] A1: Quantum Advantage

[0175] Now, we want to optimize the expected utility over different strategies. We can see from eq. (A.1 ) that for such a purpose the key object of interest is the weighted utility array:

[0176] Indeed, we can express the expected utility of a behavior p(d\o) as simply:E 44

[0177] We denote the set of all possible deterministic behaviors as D and quantum behaviors as Q. Then, we define the classical value and quantum value of w as:

[0178] We will call g(w):= q*(w) - c*( v) the gap.

[0179] Now, since deterministic strategies form a subset of quantum strategies, g(w) > 0. We will particularly be interested in cases for which g(w) > 0, in which case we call w gapped. Otherwise, g(w) = 0 and we call it gapless. We denote the set of all gappedweighted utility arrays by G. We establish some basic properties of G. \Ne in particular explore transformations of an array that preserves the gapped property.

[0180] Proposition 5. G is a cone, that is, it is closed under positive scalar multiplication.

[0181] Proof. Let> 0

[0182] Then, let

[0183] It is clear that

[0184] Similarly,

[0185] Thus,« f| < Sml SO O ' W € C.

[0186] We also define the constant array&y y whsro I.

[0187] It is clear translation by a multiple e also preserves G.

[0188] Proposition 6. Let w be a weighted utility array. Then,

[0189] In particular, ifL<. s 4- 44 4 <;

[0190] Proof. We have trivially4. "" o («■) -J-

[0191] Hence,

[0192] The second statement is immediate.

[0193] Another observation is that the ordering of observations and decision is arbitrary. Hence, the following holds.

[0194] Lemma 7. The gap of a weighted utility array is invariant under permutations of observations and decisions. That is, letting w be a weighted utility array, define the multidimensional array w whose elements are:

[0195] where 77?, oy are permutations of 0 / , Di, respectively. Then, g(v) = g(w). In particular, if w e G.=, then v e G.

[0196] Proof. This is immediate via simply relabeling observations and decisions according to Tn and o / , respectively, for all strategies.

[0197] We also state properties that do not hold regarding G.

[0198] Fact 8. G is not closed under addition.

[0199] That is, G is a cone but not a convex cone. To see this, consider the weighted utility matrix WCHSH corresponding to the CHSH game:

[0200] Here, we let oi, di e {0, 1} so that we can use logical and bitwise addition operators. Intuitively, the two parties have to make opposite decisions when both observations are 1 and have to make the same decision otherwise. It is well known that WCHSH e G. Then, define the “anti-CHSH game” weighted utility matrixas

[0201] That is, it has the opposite winning condition. By a similar argument as that of the CHSH game:

[0202] However,

[0203] This establishes Fact 8.

[0204] Furthermore, counter to intuition, scaling the weighted utility by a different positive constant for each set of observations does not always preserved gappedness. This kind of scaling can be interpreted as changing the input distribution.

[0205] Fact 9. Fact 9 is:

[0206] The example is again CHSH, but this time we will make use of the correlation form:4 4

[0207] The scaling we choose is a natural scenario in which the input distribution for each party is i.i.d, according to a Bernoulli distribution with parameter p e [0, 1] instead of the uniform distribution as is usually assumed for the CHSH game. In this case, we want to instead computeHA" d AisMd 4 di - 4 £1 - "" d£'h£h£

[0208] For conciseness, we will denote the correlations with a, b, c, d E [-1, 1 ], respectively. Then, the quantum value is given by(A.3subject to•arc- a « resin & arcsin e •••■ arose d) a (A.4) and its possible permutations of a, b, c, d. We will compute this in full generality. First, we see that if eq. (A.4) is strictly satisfied, we can always increase a, b, c and decrease d until« ■■■ araia 4 and the expression in eq. (A.3) can only increase. Thus, it is sufficient to consider the equality condition eq. (A.5). We will also see that this is sufficient to satisfy all other permuted versions of eq. (A.4), so the result must be the maximum. We make use of the method of Lagrange multipliers:£ d 4' d I pd 4 p(i "" 4?; "" -Ad - A s a s '< > S; < a 4- mead 4 sumajs < • arced 4 -

[0209] We solve for the stationary points:™ (1.?>£iAE V 1 —so

[0210] Similarly, / A'-. / ,.v / A~S ™::fc < / 1 "" ~v~ - A < i 1 4:a I "' — r ■y p-U - pH y p~u - pp ' y p!

[0211] In particular, we see that b = ±c. If b = -c, the correlation expression becomes:(I p4'« ■■■■ < (1 -;>}■ 4-?r,

[0212] The RHS can be obtained by the classical behavior where a = 1, b = 1, c = -1, d - -1, which also does not attain the classical value (It is clear that depending on the value of p, we either want to set either the first or last term in eq. (A.3) to be negative.). Thus, WOLOG, we will set c to equal b. Thus, we have the simpler optimization:whereC: ( I — pfe 4- 1 p 4 -■ (A.t?) and;<< 4... asesAs 4 s™?r. ( A 7)

[0213] Let| 1 x < 0ppp s ssss 4 i s;■:::• {>(1.4 > 0be the sign function.

[0214] We consider the following cases:

[0215] 1. sgn(a) sgn(d) > 0: Then, taking the cosine of both sides ofas.-a areai rs 4 4' -™? areasrs Awe obtainsince cosine is an even function. We simplify this equation to get

[0216] We plug this into Mathematica and obtain

[0217] The A = 0 solutions are classically attainable solutions, which might be optimal for certain values of p.

[0218] 2. sgn(a) sgn(d) < 0: We follow similar steps to get

[0219] We need to check what conditions on p G [0, 1] guarantee that

[0220] We see that this is true if

[0221] We can easily check that. Fand so is'V pl* >their geometric mean. This ensures thatwhich means when- ~ R.;we obtain feasible solutions.

[0222] Now, we observe that to satisfy eq. (A.7), we must have at least one of a, b > 0. Furthermore, at least one of b, d > 0. We therefore analyze 5 cases. We will see that in every case where ± * is feasible,iH'cskss* < sit ess u 6, (AJJj

[0223] Since 2(arcsin x - arcsin y) > -2zTfor all x, y and eq. (A.5) holds, all permutations of eq. (A.4) are satisfied.

[0224] 1. a, b, d > 0: Then sgn(a) sgn(d) > 0. Hence,which is feasible when€ (1.,by checking if eq. (A.7) is satisfied. Note that the three-term expression is written in the same order as eq. (A.6). In this range for p, we can further simplify this toNote that sinceAs a, b, dare all non-negative, eq. (A.9) holds. There is also the classical solution which is possible for all p G [0, 1]:

[0225] 2. a, b > 0, d < 0:C[. V:, A,. V ■ vSl ■■■■ I ■■■• pjLfeasible when< •> x” < j '***".7..... *Due to the signs of a, b, dwe clearly have eq. (A.9).

[0226] 3. a, d > 0, b < 0: One can check the A* solution is not feasible for anyp e U.. X. X;The classical solution in this case is™ 444- L

[0227] 4. b, d > 0, a < 0: One can check the A* solution is not feasible for any

[0228] 5. b > 0, a, d < 0:. 2H1.?<feasible whenSinceDue to the signs of a, b, d, we again can conclude eq. (A.9). The classical solution isCv.4 ™.2 / 4 L

[0229] We combine the above results and compare the A = ±A* and A = 0 solutions and find the former solutions do better forOutside of this range the quantum value equals the classical value, the latter of which can be easily computed. Since the CHSH inequality with an independent Bernoulli distributed input is a natural generalization of the original inequality, we summarize our result as a theorem.

[0230] Theorem 10. Consider the utility array WCHSH,Pfor the CHSH game with each input distribution being an independent Bernoulli random variable with probability p. Then, the classical value is given bywhile the quantum value is given byIn particular, WCHSH,PG GwhenHence, forwe obtain Fact 9. For convenience, we also provide the numerical values for the bounds:

[0231] We can check that indeed for p = 2,as expected.

[0232] A2: XOR Arrays

[0233] An interesting class of TC problems have utilities that only depend on the XOR of the decisions. We make the following definition.

[0234] Definition 11. Suppose we have a TC problem whereAn array m indexed byis an XOR array if

[0235] We will call TC problems whose utility array is an XOR array an XOR problem. Note that this implies the weighted utility array is also an XOR array. Such problems are equivalent to correlation expressions and the quantum values can be explicitly computed via a semidefinite program (SDP).

[0236] An interesting observation is that we can concludevia Lemma 7 applied to WCHSH and setting m be a bit flip while letting all else be equal. We can generalize this observation by making the following definition.

[0237] Definition 12. Let m be an XOR array, where

[0238] Then, definethe anti-array that has elementswheredenotes the bit flip of x.

[0239] Intuitively, if w is a nonlocal XOR game, thenis the same game with the opposite winning conditions. We can now easy establish the following.

[0240] Proposition 13. Suppose w is an XOR array. Then,

[0241] Proof. Let TTI be the bit flip permutation on {0, 1}. Then,

[0242] Hence, by Lemma 7, the conclusion follows.

[0243] Intuitively, for XOR problems, the procedure of reversing the winning conditions is equivalent to local index permutation of the utility array, thereby preserving the gap.

[0244] Theorem 14. A central result for XOR problems with two parties is the theorem by Tsirelson illustrated in FIG. 15, showing four conditions for real numbers. The setis called a quantum correlation matrix. In terms of the TC problem formalism, we can relate a quantum correlation matrix CM corresponding to the quantum behaviorvia:■: r sA j) - f ) - 2p(XQR. - W A A) ■■ * > (A. K)

[0245] Note for XOR problems the expected utility of a quantum behavioronly depends on the quantitiesHXUR - rv / ..Furthermore, we can relate the operatorsAr:with the measurement operators

[0246] We make the following definition.

[0247] Definition 15. Define a projection operator n as trivial if fl is the zero or identity operator. We define a quantum strategy to be degenerate if one party uses a trivial measurement operator. Otherwise, we call it non-degenerate.

[0248] Note that if a party with only two possible decisions uses a trivial measurement operator, they are locally implementing a deterministic strategy. We can thereby obtain the following corollary.

[0249] Corollary 16. Consider an XOR problem with two parties. Then, any possible quantum correlation matrix cw can be realized using a non-degenerate quantum strategy. Moreover, such a strategy can attain the quantum value.

[0250] Proof. Let SQbe a quantum strategy and the Cki be the quantum correlation matrix realized. If one of the parties uses a trivial measurement operator, WOLOG the first party, then for someis the zero or identity operator. Then,,4.s. - But this impliesfor any quantum state p. Thus, measurement operators obeying condition 3 of Theorem 14 must all be nontrivial measurement operators. The first conclusion follows. Since Cki are all that is involved in the expected utility of an XOR problem, the second conclusion follows.

[0251] Such results can help reduce the search space for numerical optimizers that search over all possible quantum strategies.

[0252] A3: Behaviors with Loss

[0253] Here we take a look from a theoretical standpoint the phenomenon of entanglement loss when physically implementing a quantum strategy. As mentioned in Section 4.1, such loss is common when using photons to distribute entanglement. In general, when loss occurs for a certain party, that party can fall back to a predetermined local deterministic strategy. We can rigorize this concept using the following definition 17.

[0254] Note that instead of defining a semiclassical strategy as a special case of a quantum strategy, we could define it as a generalization of a classical strategy, which would then allow for the choice of trivial measurement operators to be based on observations and shared randomness. This is more general than what we define but does not lead to higher expected utilities. Furthermore, “semiclassical” more often implies starting from quantum and then taking the classical limit.

[0255] Definition 17. LetThen, we define an S-semiclassical strategy as a quantum strategy where the measurement operators of the parties in S are all trivial for all possible observations. That is,are either the zero or identity operator.

[0256] In other words, the parties in S are each implementing a local deterministic strategy. An S-semiclassical behavior is the corresponding behavior of an S-semiclassical strategy. Now, in general entanglement loss is stochastic, therefore leading to the following definition.

[0257] Definition 18. Let Sq be a quantum strategy, Sd a deterministic strategy, andThen, we useto denote the S-semiclassical strategy obtained by modifying the quantum strategy Sqso that the each party in S locally implements the deterministic strategy Sd.

[0258] Next, let® 0 IO. n whereThen, the { / ];} / -lossy behavior of the tuple (Sq, Sd) is given bywhere ps(d\o) denotes the behavior corresponding to a strategy S.

[0259] Finally we defineas the { / ] / } / -lossy value of a TC problem the maximum expected utility with respect to all possible { / / } / -lossy behaviors.

[0260] It will be useful to define the { / j / j / '-lossy Bell operator for a tuple (Sq, Sd)Lwhere fli(Sq us Sd) are the measurement operators used in the S-semiclassical strategy Squs Sd, which is itself a quantum strategy. The largest eigenvalue of this operator is highest expected utility for the choice of measurement operators over all possible shared quantum states.

[0261] It is straightforward to show that the set of all quantum behaviors Q is convex. Since a semiclassical behavior is a quantum behavior, by the definition in eq. (A.11) we can conclude a lossy behavior also belongs to Q. Thus, all possible { / / Ji-lossy behaviors constitute a subset of Q. We can interpret { / 7 / }i as a “shrinking factor” of Q toD Q:when / i = 1 for all / , we can attain all of Q, whereas when= 0 for all / , we can only attain D.

[0262] We will establish the following basic result.

[0263] Proposition 19. Consider a (2,2,2) problem. Then, the behavior of a degenerate quantum strategy is a classical behavior. Moreover, the same is true for lossy behavior where the quantum strategy is degenerate.

[0264] Proof. The set of all possible classical behaviors is given by a polytope, called the local polytope. The inequalities that define the local polytope for the case of two parties, two observations, and two decisions are positivity conditions and permutations of the four terms in the CHSH inequality.

[0265] We will explicitly show that the behavior of a degenerate quantum strategy satisfies all possible CHSH inequalities. WOLOG, suppose the first party always outputs 0 when they make the first observation. When they make the second observation, they perform some quantum measurement {FI, / A - FIA}. Let the second party’s measurement operators be given by {FIB, / B - Fl B} and {ITB, IB - FI'B} in order of their observations. Then, we can computeOh'0,0}) - Ms?. ® p(. XGR::: DjfQ,! R?? I lf?)P K j?«XOR 0j( 10}) - MHls?Ms:dl,- Klbhd?■1MM M'MRM -• * M A RP(XGR. - oj(T 2M(n..?x ~ i ™ Mtu R

[0266] Now, the CHSH inequalities are usually expressed in terms of the quantum correlation matrix elements cw. We first consider the usual permutation

[0267] Now, it is easy to see that the operatorII.: H;-. - 11. - is an orthogonal projector. Thus, its eigenvalues are either 0 or 1. Hence,

[0268] By symmetry, we only need to consider one other permutation:n:n n-

[0269] Againn / < - 1 iis an orthogonal projector. Thus,

[0270] We can conclude that the quantum behavior satisfies the inequalities of the local polytope and is therefore a classical behavior.

[0271] Now, given a tuple (Sq, Sd) where Sqis degenerate, the corresponding lossy behavior is a convex combination of quantum behaviors (possibly semiclassical) where each behavior results from a degenerate strategy. Hence, by the previous result, the lossy behavior is a convex combination of classical behaviors and is therefore itself classical.

[0272] We next prove that 2-dimensional (qubit) quantum systems are sufficient to achieve lossy values for (n, 2, 2) problems.

[0273] Proposition 20. Consider an (n, 2, 2) problem. Then, the {r / J-lossy value can be attained where the optimal quantum strategy Sq uses only qubit quantum systems.

[0274] Proof. Fix a deterministic strategy Sd. Let Sqbe a quantum strategy. The results show that the extremal points of the set of all quantum behaviors for ( / ?, 2, 2) problems are realized by quantum strategies using only qubit quantum systems. Thus, in this setting every quantum behavior is a convex combination of qubit quantum behaviors. We therefore havewhereis the complementary set of S, p(ds\os) is the marginal probability onAk is a probability vector, and Sqk are qubit strategies (Note that by the no-signaling property the marginal probability does not depend on the observations of ^ ). Thus, for fixed Sd, the maximum expected utility is achieved by a qubit behavior. We can then maximize over Sd to obtain the desired conclusion.

[0275] A4: Noisy Quantum Behavior

[0276] We give a theoretical exposition on the effect of depolarizing noise in quantum strategies. We make the following definition.

[0277] Definition 21. Leti A-T. <, f M '•be a quantum strategy and v e [0, 1], Then, the v-noisy behavior of Sqis the behavior that corresponds to the strategy SQexcept the quantum state \ i ) is replaced by (1 -v)| + VTT, where TT is the maximally mixed state. That is, the v-noisy behavior is given by

[0278] The robustness v* of a quantum strategy whose expected utility is at least the classical value is the smallest v such that the expected utility of the v-noisy behavior is equal to the classical value.

[0279] We can expand out eq. (A.13):

[0280] We see that the effect of the noise is completely determined by the ranks of the measurement operators. Now, the fraction in the second term can be interpreted as a behavior. Indeed, we observe that it factorizes into behaviors for individual parties:

[0281] Hence, it is a classical behavior. We can therefore interpret v as a shrinking factor of Q to the set of all classical behaviors (with rational probabilities) that are factorizable, that is, where the behavior of each party is independent. Note that this set includes D, so we can always attain the classical value even for v = 1. Furthermore, this implies for a quantum strategy with an expected utility higher than the classical value, the expected utility for the v-noisy behavior is monotonically decreasing with v.

[0282] For the case of the hedging problem when there is a quantum advantage, if a quantum strategy only uses qubits and achieves the quantum value, the ranks of the measurement operators must be rank 1 according to Proposition 19. Hence, in this case.. 1000 (1.... ^>30)4^7.

[0283] We claim that in the presence of depolarizing noise, increasing the dimensions of the quantum systems used can increase the expected utility, even for the CHSH inequality. Recall the utility array is given byand the inputs are uniformly distributed. Then, there exists an optimal qubit strategy for the noiseless setting, call it SQ, which is non-degenerate. Thus, the factorizable behavior in eq. (A.14) must be the uniform distribution. The expected utility for the v-noisy behavior of Sqis' > s ■ '■ 2

[0284] Now, consider modified quantum strategy Tqwhere both parties use ququarts (dimension 4). Instead of the original entangled state \ j}, we useAnd for the all the projectorsfor the 0 decision, we replace it with(the corresponding 1 decision projector is the orthogonal complement). This clearly preserves the expected utility of the noiseless quantum behavior. However, the ranks of the projectors are not doubled while the quantum dimensions are. It is not difficult to see that the factorizable behavior in this case is, in matrix form,O (O L0 L Iwhich implies for Tq, the v-noisy behavior achieves an expected utility of..?r 9, Ifor v > 0. This establishes the claim.

[0285] Appendix B: Numerical Optimizer

[0286] We give some details of the numerical optimizer used to compute classical and quantum values, as well as how some of the figures were computed. The optimizer itself is quite straightforward but can be used for arbitrary TC problems and not just special classes such as XOR problems.

[0287] B.1 Classical value

[0288] The classical value is straightforward to compute. The optimizer proceeds via brute force by iterating over all possible deterministic strategies and tracking the largest expected utility found. Each deterministic strategy is simply a choice of decision given an observation for each party, which for aTC problem, would simply be parameterized bydiscrete variables, the variables for party i taking |D| different values. Aside from this brute force approach, there are also linear programming methods that can be used.

[0289] B.2 Quantum value

[0290] Our numerical optimizer computes the quantum value via an optimization over explicit parameterizations of projective measurements on a q-dimensional Hilbert space Hqwith A < q possible outcomes. This parameterization has both continuous and discrete variables.

[0291] The continuous variables parameterize an orthogonal basis of Hqmodulo nonzero scalar multiplication.

[0292] A parameterization of a unitary matrix via q2real parameterswhere:and where the matrix multiplication proceeds from left to right,and

[0293] The range for Am,n is [0, 2TT] for m > n and [0, 11 / 2] for m < n. Since an orthogonal basis modulo nonzero scalar multiplication is basically a unitary matrix modulo multiplication by a diagonal unitary, we see from eq. (B.1) that we simply can remove the rightmost term to obtain our desired parameterization. Thus, in the end we only need the q2- q parameters Am,n where ml= n, and use the parameterizationwhere the columns of U are the basis vectors that we use.

[0294] We have figured out how to parameterize a projective measurement for each party. We next must combine the parameters for each party together to parameterize a Bell operator in eq. (A.2). Note that we do not need to parameterize the shared quantum state since we can simply compute the largest eigenvalue of the Bell operator. In fact, WOLOG we can assume each party performs the measurement in thecomputational basis upon the first observation, with respect to some ordering of the observations. This is because we can always conjugate the Bell operator by a tensor product unitary that rotates for each party the measurement operator used upon making the first observation to the computational basis. In sum, for a TC problema quantum strategy where party / uses a quantum system of dimension q, has a total ofcontinuous variables anddiscrete variables.

[0295] Interestingly, this is not always the minimum number of variables we need: for some cases we can further eliminate variables. In the case of (n, 2, 2) problems, we can obtain the following result.

[0296] Proposition 22. Consider an (n, 2, 2) problem. Then, it is sufficient to use n continuous variables to parameterize a quantum strategy that achieves the quantum value. Moreover, it is also sufficient to parameterize a quantum strategy that with the appropriate deterministic strategy achieves the lossy value.

[0297] Proof. By Proposition 20, to achieve the quantum value we can assume we are only using qubit systems. By eq. (B.3), we therefore seem to need; 2 ■ (2 ■■■■ h 2»continuous variables to parameterize a Bell operator up to local unitary equivalence. We can halve this. LetX U?denote the continuous parameters for party j’s projective measurement upon their second observation (recall measurement upon first observation is in the computational basis). Now, for q = 2, eq. (B.2) gives

[0298] Let |v) denote the first column vector. Hence, considering all possible observations and partitions, the choices of a projector for party j is the following:n. {MHU - • idoHC J s) CH }. (iu

[0299] Then, we observe for all possibilities, the operatoris independent of

[0300] Here we used the Z rotation matrixv. A0\<. t \n

[0301] Thus, we see thatis independent of; / :for all j. However, this operator is local unitarily equivalent to the Bell operator. This implies that the largest eigenvalue is independent of these n parameters and so the first conclusion follows.

[0302] In the lossy case, for any quantum strategy Sqand deterministic strategy Sd, every term in the convex combination of eq. (A.12) also satisfies the above independence property since eq. (B.4) still holds. The second conclusion therefore follows.

[0303] As an example, a quantum strategy for a (2, 2, 2) problem using only qubits would have 2 continuous variables and 2 + 2 = 4 discrete variables. Note that by Proposition 19 in this case a quantum strategy whose behavior is not classical uses only nontrivial measurement operators. Thus, the 4 discrete variables can be eliminated as we only need to consider the 1 + 1 = 2 partition. Thus, to compute the plots previously shown above, we need to only optimize over 2 continuous variables.

[0304] Optimization is conducted via two methods. The first is brute force. For the continuous variables, we choose a grid of values to evaluate over and also perform a gradient descent for each point in the grid. This is implemented via a program called scipy.optimize. The discrete variables are evaluated at every possible value. This method is clearly not scalable, so we also include another method, the CMA-ES evolutionary algorithm with discrete variables. We use the Python implementation in our optimizer. Note that CMA-ES does not guarantee a global optimum. However, in practical scenarios it may be sufficient to find a quantum strategy with a higher expected utility than the classical value. To double-check the results in plots provided, we performed both brute force (with a grid linear size of 20 points) and CMA-ES optimizations and verified that we obtained the same results. Note that since the hedging problem is an XOR problem, we can also use semidefinite programming to solve for the quantum value. We instead use our optimizer since it can handle general TC problems and use the hedging problem as an example of how to apply it.

[0305] B3: Lossy value

[0306] To compute the lossy value of a TC problem, we optimize over tuples (Sq, Sd). This simply combines the parameterization of a deterministic strategy and that of a quantum strategy. The number of continuous variables is the same, but now the number of discrete variables is25"

[0307] Note that we are now instead computing the largest eigenvalue of the lossy Bell operator in eq. (A.12), so we again do not need to parameterize the shared quantum state. We can perform optimization via brute force where for the deterministic strategy variables we also iterate over all possible values. The CMA-ES optimizationcan be done by treating the deterministic strategy variables as just additional discrete variables.

[0308] By Proposition 19 and Proposition 20, we can compute the lossy value of a (2, 2, 2) problem by parameterizing a quantum strategy that only uses qubits and nontrivial measurement operators. By Proposition 22, we again only need 2 continuous variables. However, we now have 4 discrete variables parameterizing the deterministic strategy which must be included in the optimization. We perform brute force optimization over both continuous (grid linear size of 20) and discrete variables for a given ri to compute the lossy value, then perform a binary search over n e [2 / s, 1] to find / ]*, thereby obtaining the results disclosed herein.

[0309] Additional advantages and modifications will readily occur to those skilled in the art. Therefore, the invention in its broader aspects is not limited to the specific details and illustrative examples shown and described herein. Accordingly, various modifications may be made without departing from the spirit or scope of the general inventive concept as defined by the appended claims and their equivalents.

Claims

What is claimed is:

1. A system for tacit coordination of decisions, the system comprising:a device for generating entangled photon pairs having a measurable state; a first measurement device;a second measurement device;a first quantum channel photonically connecting the device for generating the entangled photon pairs and the first measurement device; and a second quantum channel photonically connecting the device for generating the entangled photon pairs and the second measurement device; wherein the first measurement device is configured to measure the measurable state of first photons of the entangled photon pairs generated by the device for generating the entangled photon pairs that are successfully received by the first measurement device based in part on a measurement setting determined using first local information and thereby obtain a first measurement result,wherein the second measurement device is configured to measure the measurable state of second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs that are successfully received by the second measurement device based in part on a measurement setting determined using second local information and thereby obtain a second measurement result, wherein, to make tacitly coordinated decisions that achieve a quantum advantage,the first measurement device is configured to make a first local decision, said first local decision being made using a predetermined quantum strategy based on the first measurement result when the first photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are successfully received by the first measurement device, and said first local decision being made using a predetermined deterministic strategy when the first photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are not successfullyreceived by the first measurement device due to photon loss in the first quantum channel, andthe second measurement device is configured to make a second local decision, said second local decision being made using the predetermined quantum strategy based on the second measurement result when the second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are successfully received by the second measurement device, and said second local decision being made using the predetermined deterministic strategy when the second photons of the entangled photon pairs generated by the device for generating the entangled photon pairs are not successfully received by the second measurement device due to photon loss in the second quantum channel, andwherein the predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account.

2. The system according to claim 1, wherein the first measurement device is located in a colocated server of a first stock exchange, and wherein the first local information comprises a real-time market data feed from the first stock exchange.

3. The system according to claim 2, wherein the second measurement device is located in a colocated server of a second stock exchange, and wherein the second local information comprises a real-time market data feed from the second stock exchange.

4. The system according to claim 3, wherein the tacitly coordinated decisions are trade decisions executed at the respective first and / or second stock exchanges.

5. The system according to claim 1, wherein the first and / or second quantum channels are selected from the group consisting of vacuum beam guides, optical fibers, free space, or optical waveguides.

6. The system according to claim 1, wherein the measurable state of the entangled photon pairs is that their polarization degrees of freedom are entangled.

7. The system according to claim 6, wherein the first measurement device comprises a waveplate and a polarizing beam splitter that routes photons with different polarization to different respective photon detectors.

8. The system according to claim 7, wherein the waveplate is tunable based on the first local information.

9. The system according to claim 7, wherein the tacitly coordinated decision made by the first measurement device is based upon which of the different respective photon detectors measured the measurable state of an entangled photon of the entangled photon pairs.

10. The system according to claim 1, wherein the measurable state of the entangled photon pairs is encoded in their spatial modes.

11. The system according to claim 10, wherein the first measurement device is configured to convert spatial mode encoding to polarization, and wherein the first measurement device comprises a waveplate and a polarizing beam splitter that routes photons with different polarization to different respective photon detectors.

12. The system according to claim 1, wherein the device for generating the entangled photon pairs is configured to perform spatial mode encoding with a fixed linear optical module.

13. The system according to claim 1, wherein the device for generating the entangled photon pairs is configured to perform encoding of cluster states via electrons emitting photons.

14. The system according to claim 1, wherein the device for generating the entangled photon pairs is configured to perform bosonic quantum encoding that is robust against photon loss errors.

15. The system according to claim 14, wherein the system is configured to use bosonic quantum error correction to suppress photon loss errors of up to 50% loss to deterministically generate entanglement.

16. The system according to claim 15, wherein quantum error correction is performed before the measurement to suppress loss error.

17. The system according to claim 1, wherein the device for generating the entangled photon pair comprises an optically pumped nonlinear crystal.

18. The system according to claim 1, wherein the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first server and a second server, respectively, that are spaced apart such that their decisions have to be made after their observations within a time less than their communication latency.

19. The system according to claim 18, wherein the system further comprises a preprocessor that divides a data stream into a first fraction that comprises the first local information and a second fraction that comprises the second local information.

20. The system according to claim 1, wherein the first measurement device and the second measurement device are configured provide tacitly coordinated local decisions to a first processor and a second processor, respectively, at a time scale less than their communication latency.

21. The system according to claim 20, wherein the first processor and the second processor have a shared memory and the system minimizes address collisions.

22. The system according to claim 1, wherein the device for generating the entangled photon pair is configured to perform time-bin encoding with fast linear optics.

23. A system for making tacitly coordinated decisions, the system comprising: a first measurement device;a second measurement device;a first quantum memory device coupled to the first measurement device;a second quantum memory device coupled to the second measurement device; a heralded entanglement measurement device;a first quantum channel photonically connected between the heralded entanglement measurement device and the first quantum memory device; anda second quantum channel photonically connected between the heralded entanglement measurement device and the second quantum memory device;wherein the first and the second quantum memory devices are each configured to repeatedly emit photons toward the heralded entanglement device via the first and second quantum channels,wherein the heralded entanglement device is configured to detect photon loss errors,wherein the heralded entanglement measurement device is configured to transmit heralding signals to the first and second quantum memory devices, respectively, indicating when the first and second quantum memory devices have been successfully entangled,wherein the first measurement device is configured to perform a measurement of the first quantum memory device based in part on a measurement setting determined using first local information,wherein the second measurement device is configured to perform a measurement on the second quantum memory device based in part of a measurement setting determined using second local information, andwherein, the first and second measurement devices are configured to make the tacitly coordinated decisions based on the measurements made by first and second measurement devices, thereby obtaining a quantum advantage.

24. The system according to claim 23, wherein the first and second measurement devices are configured to make the tacitly coordinated decisions using a predetermined optimal quantum strategy when they determine that there are sufficient entangled memory devices to do so, and are configured to make the tacitly coordinated decisions using a predetermined optimal deterministic strategy when they determine that there are insufficient entangled memory devices to do so.

25. The system according to claim 23, wherein, to suppress depolarizing errors, the system is configured to use an entanglement distillation procedure, which uses multiple Bell pairs and local operation and classical communication to distill Bell pairs with higher fidelity.

26. The system according to claim 25, wherein the distilled Bell pairs are used for teleported gates to obtain logical Bell pairs with quantum error correction protection.

27. The system according to claim 23, further comprising:a first quantum transducer coupled to the first quantum memory device; and a second quantum transducer coupled to the second quantum memory device; wherein the first and second quantum transducers are configured to convert the frequencies of the photons emitted by the entangled quantum memory devices, while preserving entanglement, for transmission through the first and second quantum channels, respectively.

28. The system according to claim 23, wherein the first measurement device is located in a colocated server of a first stock exchange, and wherein the first local information comprises a real-time market data feed from the first stock exchange.

29. The system according to claim 28, wherein the second measurement device is located in a colocated server of a second stock exchange, and wherein the second local information comprises a real-time market data feed from the second stock exchange.

30. The system according to claim 29, wherein the tacitly coordinated decisions are trade decisions executed at the respective first and / or second stock exchanges.

31. The system according to claim 23, wherein the first and / or second quantum channels are selected from the group consisting of vacuum beam guides, optical fibers, free space, or optical waveguides.

32. The system according to claim 23, wherein the system comprises first and second lasers configured to drive the first and the second quantum memory devices, respectively, to emit the photons to the heralded entanglement device via the first and second quantum channels.

33. The system according to claim 32, wherein the system further comprises a first and second microwave drives configured to perform quantum gate operations on the first and second of the pair of quantum memory devices based on local information when the heralding signal indicates that the quantum memory devices have been successfully entangled.

34. The system according to claim 33, wherein the system further comprises first and second measurement lasers directed at the first and second quantum memory devices that generate state dependent fluorescence which the first and second measurement devices are configured to detect to make the tacitly coordinated decisions.

35. The system according to claim 34, wherein the first and second measurement devices each comprise a camera for detecting the state dependent fluorescence.

36. The system according to claim 23, wherein the first and second quantum memory devices comprise at least one of:atoms;ions;superconducting qubits;nitrogen vacancy centers;nuclear / electron spins; andmicrowave cavities.

37. A method for making tacitly coordinated decisions given first and second local information, respectively, the method comprising:emitting a pair of photons, said pair of photons including a first photon emitted from a first quantum memory and a second photon emitted from a second quantum memory;measuring the pair of emitted photons to determine whether the first and second quantum memories are entangled;when the first and second quantum memories are determined to be entangled, sending heralding signals to the first and second quantum memories that indicates successful or unsuccessful entanglement;storing the successful entanglement in the first and second quantum memories, respectively;performing quantum operations on the first and second quantum memories given the first and second local information;consuming the stored successful entanglement by measuring the first and second quantum memories; andmaking the tacitly coordinated decisions based on the measurements of the first and second quantum memories to obtain a quantum advantage.

38. The method according to claim 37, wherein the first local information comprises a real-time market data feed from a first stock exchange, and the second local information comprises a real-time market data feed from a second stock exchange.

39. The method according to claim 38, wherein the tacitly coordinated decisions are made faster than latency in communication between the first stock exchange and the second stock exchange.

40. The method according to claim 39, wherein the real-time market data feed from the first stock exchange and the real-time market data feed from the second stock exchange include data that indicates a statistical relationship between a plurality of financial instruments.

41. The method according to claim 40, wherein the tacitly coordinated decisions are trading decisions.

42. The method according to claim 41, wherein the trading decisions are used for hedging.

43. The method according to claim 37, wherein the tacitly coordinated decisions are computations made by a plurality of servers performing distributed computing.

44. The method according to claim 37, wherein the tacitly coordinated decisions are instructions executed by separate processors.

45. The method according to claim 44, wherein the separate processors read from a shared memory and the tacitly coordinated decisions minimize address collisions.

46. The method according to claim 45, wherein the first and second local information comprise addresses in the shared memory that the separate processors are to query.

47. The method according to claim 37, wherein a determination is made whether depolarizing noise from memory-photon entanglement, imperfect photon detection, decoherence of the first and second memory devices during photon traversal and / or measurement of the first and second memory devices exceeds a predetermined robustness value such that the tacitly coordinated decisions made at the predefined frequency obtain the quantum advantage.

48. A method for making tacitly coordination of decisions that achieve a quantum advantage, the method comprising:generating entangled photon pairs having a measurable state;transmitting, via first and second quantum channels, respectively, first photons of the entangled photon pairs to a first measurement device and second photons of the entangled photon pairs to a second measurement device; measuring, using the first measurement device, the measurable state of the first photons of the entangled photon pairs that are successfully received by the first measurement device based in part on a measurement setting determined using first local information to thereby obtain a first measurement result;measuring, using the second measurement device, the measurable state of the second photons of the entangled photon pairs that are successfully received by the second measurement device based in part on a measurement setting determined using second local information to thereby obtain a second measurement result;to achieve the quantum advantage, making a first local decision via the first measurement device using a predetermined quantum strategy based on the first measurement result when the first photons of the entangledphoton pairs are successfully received by the first measurement device and using a predetermined deterministic strategy when the first photons of the entangled photon pairs are not successfully received by the first measurement device due to photon loss in the first quantum channel, and making a second local decision via the second measurement device using the predetermined quantum strategy based on the second measurement result when the second photons of the entangled photon pairs are successfully received by the second measurement device and using the predetermined deterministic strategy when the second photons of the entangled photon pairs are not successfully received by the second measurement device due to photon loss in the first quantum channel, wherein the predetermined quantum strategy and the predetermined deterministic strategy both take photon loss error rate into account.

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