In this article I will discuss a brief introduction to the basics of performing quantum computations with single photons.

We’ve all heard that **light can behave as both a wave and a particle**, and more than once I have personally ended up with a headache while my mind tried to wrap around the particle-wave duality. When you see a rainbow (or Pink Floyd’s Dark Side of The Moon album cover), you’re basically watching how light waves bend (refract) as they enter a different medium at an angle. But as Einstein proved in 1905 by explaining the photoelectric effect, light can also behave as a particle. Einstein proposed that **photons are the basic elementary constituents of light** which contain the energy corresponding to their wavelength.

Thus, **photons are quantum systems that can be seen as energy packets traveling thought space at the speed it light**.

In order to further illustrate how light can behave as a wave and as a particle in an optical circuit, let us consider an experiment in which **light is sent through a 50/50 beamsplitter (BS)**.

A classical pulse of light pulse of intensity is **split into two independent beams** of intensity .

However, when a single photon passes through a beamsplitter, something completely different happens. Just like Schrödinger’s cat, t**he resulting quantum state for the photon is a superposition** of the photon traveling trough one path or the other. The final state of the single photon features **Quantum Correlations** that can be used in Quantum Information schemes [1].

As discusses in this article, in order to perform information processing tasks with a given quantum system, one must be able to:

- Prepare quantum states in a given initial qubit state with longer decoherence times.
- Control with a high precision the quantum states (i.e., being able to perform quantum gate operations).
- Control two-qubit interactions.
- Measure qubit states.

Although these conditions are not necessarily universal, they will be helpfully to guide the present article and understand quantum computation with light.

It’s easy to think that optical schemes employ lasers to produce quantum states, but the reality is that laser pulses can contain an extremely large number of photons. For instance, a laser pulse of 1 J at 532 nm contains in average photons! [2]

Even an extremely attenuated pulse can posses (on average) more than one photon. Hence, in the past few decades a considerable amount of effort has been put forward towards developing optical devices which can precisely produce one photon at a time.

One such way of preparing single photons is through isolates two-level **atom-like systems** such as atoms, quantum dots or color centers [3]. Here, the** system is excited so that a single photon is emitted when it decays back to it’s ground state**.

Another proposal to create single photon states is through a process known as **Spontaneous Parametric Downconversion** (SPDC), where a pumped laser pulse passes through a nonlinear medium in such a way that there is a chance for correlated pairs of photons occasionally to emerge. This phenomenon can be understood in a quantum-mechanical framework as a process in which a single pump photon is annihilated and two lower energy down-conversion photons are created, under the

conditions that energy and momentum are conserved. **Two different types of SPDC can be distinguished**: when the photons have the same polarization which is

orthogonal to the polarization of the pump laser (Type-I), or when they have orthogonal polarizations (Type-II).

It is worth noting that the probability that an SPDC process takes place is of the order of per photon passing through the crystal, but since a pumped laser pulse contains an enormous amount of photons, **the resulting probability is high enough for experiments to be feasible**. I’d also like to mention that more than two photons can be created in a SPDC, but the probability of such an event can be neglected (even when considering all the photons that interact with the crystal per unit of time).

So, **how can this process to prepare single photons states for our circuit**? Well, we “*simply” *[4] have to create an optical scheme so that when a pair of photons is created one heads towards a detector and the other towards an optical fiber. That way, when the detector *clicks*, we know that a single photon has entered the fiber.

Let’s consider now **how to encode information in a single photon quantum state**. In other words, this means that we have to determine in **which degrees of freedom we are going to realize the qubit**. One possibility is to send the photon through a BS and use it’s **path to encode information**. Here, the states and correspond to the photon being on one path or the other.

One could also e**ncode information in the polarization degree of freedom** of the photon. Thus, a photonic qubit can be defined as

, ,

where represents the horizontal polarization and the vertical polarization. Similarly, we could define different basis for our photonic qubit, like the diagonal polarization and the circular polarization basis .

Now that we have understood how to codify information in our photonic state, **we still need to be able to manipulate it**. Luckily, this problem has already been solved by physicists and engineers who have worked with classical light. For instance, **the polarization of a photon can be controlled with wave plates,** and by combining different wave plates we can perform single-qubit gate operations.

In particular, by means of a half-wave plate (HWP)

and a quarter-wave plate (QWP)

we can perform any quantum operation on the photon qubit state:

Note that we can convert** between polarization and path encoding by using a Polarizing Beamsplitter **(which split light by polarization state rather than by intensity).

As a final remark for this section, I’d like to mention that there are also other schemes to encode information in different photonic degrees of freedom, like the single rail encoding or the temporal profile encoding, but I will not describe those in this article.

Let us tackle first the problem of **producing entangled photons**. As shown in Fig. 3, a Type II SPDC process produces two photons with orthogonal polarizations ( and ) that are **constrained to be within two cones**, whose axes are symmetrically arranged relative to the pump beam [7]. If the axes of the cones are such that the cones intersect,** then photons emitted into the directions of the lines of intersection cannot be assigned to one of the cones.** This process results in a **pair of entangled photon whose polarizations are perpendicular**:^{}

.

Thus far we know how to prepare single photon states, entangled states and how to manipulate each photon individually. However, we know that in order to achieve a **set of universal quantum gate**s (i.e., a set of gates to which any operation possible on a quantum computer can be reduced) **we need two-qubit entangling gates**.

This requisite is met quite easily in quantum computing schemes based on interacting systems such as superconducting Josephson junctions, cold atoms in optical lattices, trapped ions and quantum dots. Unfortunately, **photons only interact weekly with their environment and with each other**. This feature may allow us to transmit quantum photonic states over long distances with low decoherences, but at the same time it makes the task of creating two-photon gates a difficult one.

Since two-photon operations are inherently nonlinear, and up to the this date there are **two different proposals to perform entangling logic gates**: using **nonlinear optical components**, and **using ancilla photons and postselection** to provide **measurement-induced nonlinearities** [6,8,9].

The first approach uses optical nonlinearities such as:

**The Optical Kerr Effect**, which is a change in the refractive index of a material in response to an applied electric field (in this case, the laser itself). Its physical origin is a**nonlinear polarization generated in the medium, which itself modifies the propagation properties of the light**[10]. Unfortunately, this effect con only be significantly observed in experiments involving very intense beams such as those from lasers.**The Zeno effect**, which describes how the evolution of quantum systems can be slowed-down or even inhibited by repeated measurements. In this set-up, the two photonic qubits are guided within two fiber-cores doped with two-photon

absorbing atoms, so that**unwanted two-photon amplitudes are suppressed due to****the quantum Zeno effect**(note that the absorption of a photon is in this case equivalent to measuring the position of the absorbing atom) [11]. This approach is, however, usually accompanied by too much photon loss leading to the failure of the quantum gate.

In the second approach, **ancilla photons are ****combined with the logical qubits by means of beam splitters and phase shifters.** Then, the ancilla photonic qubits are measured so that we know that the correct logical output was produced if the measurements yield certain results. The output can be corrected for other measurement results.

Even if in the first approaches we have to deal with photon losses and small nonlinearities, **their main advantage over the second ones is that they require much less resources when compared to linear optics measurement-induced nonlinearities.**

By now, the only ingredient we are missing to perform quantum computations with photonic devices is the ability to measure single photon states. Ideally, a single-photon detector should have [12]:

- A
**high detection efficiency**, i.e., a high probability that a photon incident upon the detector is successfully detected, - A
**zero dark-count rate**, meaning that the detector will not output pulses in the absence of any incident photons, - A
**zero dead time**, which entails that no incident photons will be lost between photon-detection event, - The
**capacity to distinguish the number of photons in an incident pulse**(many detectors work as photomultiplier, meaning that they can’t resolve the number of photons, they can only tell if a photon has been detected or not).

Although** we are still a long way from achieving an ideal single-photon detector which meets all the previous requirements**, recent advances in **state-of-the-art technologies yield promising results** in both non-photon-number-resolving single-photon detectors and in photon-number resolving detectors [12].

*Until then, we will have to make do with what we have.*

**Linear optical devices have provided us with an attractive scenario to study and realize quantum information processing tasks**. Nowadays they are used for a wide range of applications like the simulation of quantum systems, cryptography, transmission and teleportation of information, quantum key distribution, quantum metrology and as hardware to perform quantum algorithms.

However, s**everal problems still have to be overcome before a large-scale optical quantum computer is to be realized**. Most optical devices are not scalable due to their single photon sources and large amount of optical elements, while at the same time more efficient methods for the creation, manipulation and detection of single photonic qubits will have to be developed.

Various devices are currently being developed to overcome such problems, like **hybrid systems**, which combine photonic qubits with quantum systems which easily enable multi-qubit interactions; and **on-chip silicon photonic technologies** [13] which are easy to fabricate and are therefore promising to be cost-effective.

[1] Single Photons for Quantum Information Processing. D. Fattal, PhD Thesis.

[2] https://mikemurphyblog.com/2016/07/04/how-many-photons-in-a-laser-pulse/

[3] Quantum Information Processing with Single Photons. Y. L. Lim, PhD Thesis (2005).

[4] Generally, the down-conversion photons are not in the visible range, and the alignment process of the optical devices can be a tough and time consuming task! *Thank you experimental physicist, we love you! *

[5] Optical Quantum Computing, J. L. O’Brien, Science 318, 1567–70 (2007).

[6] Quantum computing with photons: introduction to the circuit model, the one-way quantum computer, and the fundamental principles of photonic experiments. S. Barz, *J. Phys. B: At. Mol. Opt. Phys.* **48** 083001 (2015).

[7] Spontaneous Parametric Downconversion wikipedia entry.

[8] Quantum Computing using Linear Optics. T.B. Pittman, B.C. Jacobs and J.D. Franson, Johns Hopkins APL Technical Digest Vol. **25**, 84-90 (2004).

[9] A scheme for efficient quantum computation with linear optics. E. Knill, R. Laflamme and G. J. Milburn. Nature **409**, 46–52 (2001).

[10] Kerr effect.

[11] Entangling photons via the double quantum Zeno effect. N. ten Brinke, A. Osterloh, and . Schützhold, Phys. Rev. A **84**, 022317 (2011).

[12] Invited Review Article: Single-photon sources and detectors. M. D. Eisamana, J. Fan, A. Migdall, and S. V. Polyakov, Review of Scientific Instruments **82**, 071101 (2011).

[13] https://www.nature.com/nphoton/focus/siliconphotonics/index.html

Quantum Computation with Light por Marco Vinicio Sebastian Cerezo de la Roca se distribuye bajo una Licencia Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional.

Basada en una obra en https://entangledphysics.com/.

Exact ground states of strongly interacting spin systems are normally highly entangled and complex. Yet at particular finite values of the applied magnetic field such systems may exhibit a completely separable exact ground state, i.e. a simple product state where each spin points along a specific direction. Such remarkable phenomenon is known as ground state factorization.

Here we show the existence of an exceptionally degenerate set of completely separable ground states in XXZ systems immersed in a nonuniform field, arising at a remarkable quantum critical point where all magnetization plateaus merge. These points can emerge in arrays of any size, spin and dimension for a wide range of nonuniform field configurations, and exhibit other special critical properties like long range pair entanglement in their immediate vicinity.

The XXZ model is an archetypal quantum spin system which has helped to understand interacting many-body systems and their quantum phase transitions, and which has recently received renewed attention due the possibility of its finite size simulation with state-of-the-art technologies and its potential for implementing quantum information processing tasks. Among other possibilities, present results open the way to design exactly separable ground states in such system through a finite controlled field, which can be used as initial states for quantum protocols.

You can find the paper here:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.220605

And on the arXiv:

https://arxiv.org/abs/1708.00565

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All image credits go to Zach Weinersmith’s Saturday Morning Breakfast Cereal @SMBC

If sharing this image goes against any copyrights, please let me know!

]]>*Now the shepherds met together, according to custom, that they might send their monthly report about the flocks to the king; into their assembly he came having the ring on his finger, and as he was sitting among them he chanced to turn the collect of the ring inside his hand, when instantly he became invisible to the rest of the company and they began to speak of him as if he were no longer present. He was astonished at this, and again touching the ring he turned the collet outwards and reappeared; he made several trials of the ring, and always with the same result-when he turned the collet inwards he became invisible, when outwards he reappeared. *

*Whereupon he contrived to be chosen one of the messengers who were sent to the court; where as soon as he arrived he seduced the queen, and with her help conspired against the king and slew him, and took the kingdom.*

– The Ring of Gyges, Plato [1]

What would you do in the place of Gyges upon finding a ring that would turn you invisible? Would you use it for a greater good? Or for your own personal benefit? What can we expect from humans to do with power over others?

These are some of the questions about human nature arising from Plato’s thought experiment in the Republic, and they remains as valid today as they were 2,400 years ago. The ring of Gyges has been used as a paradigmatic example when discussing ethics and it can be the perfect starting point towards a discussion about the ethics of quantum computation.

Although it is still not clear what the full extent of the power of quantum computers might be, there are some very particular examples of quantum algorithms that can largely outperform their classical counterparts. One of them is Peter Shor’s factorization algorithm [2], which can factorize any number exponentially faster than any classical algorithm on a classical computer.

Factorizing a number entails the determination of all prime numbers which multiply together to result in the original number. For instance:

The possibility to factorize large numbers efficiently might seem as a handy mathematics trick your nerdy friend might want to pull out as a closing number for his one man show, but its implications are far more complex. One of the most widely used cryptography systems is the RSA public key crystosystem, which is used to encrypt your emails, your bank transactions and your credit cards numbers via the internet. The security of the messages encoded via this encryption method relies on the fact that factorizing a large number is an extremely difficult task to accomplish using your everyday computers. This means that given a sufficiently large quantum computer (i.e., with enough qubits), you could easily break the RSA public key crystosystem and decipher the contents of its message.

I’m hopping that by now you are starting to realize as I do that a quantum computer is just as powerful as the ring of Gyges, and that we should start discussing the ethical challenges that this new era of quantum information processing is producing.

It baffles me that so little is being said about the ethics of quantum computers, while the so-called “threat” of quantum computing has been so widely broadcasted [3-6]. Almost every major nation in the world acknowledges that we could be heading towards a major internet security crisis if a hacker gets its hands on a quantum computer. Most of those governments have joined Google, D-Wave, Microsoft, IBM, Toshiba, any many other private companies in the race towards a quantum computer but *no one* has raised the issue of *ethi**cs* so far.

It is quite obvious that whoever wins the “race” and has the ability to harness the power of quantum computers will hold an incredible advantage over those who don’t have access to these new technologies. The issue of cryptography is only one of many that arise when considering that a scalable quantum computer might be developed in the nearby future. How do we handle the intellectual property of scientific discoveries? What could happen if a government acquires a quantum computer and has the ability to decipher private messages? What if a company manages to construct one? Who could they sell it to? Won’t their profits margins bias their decisions? Wont’ they use their ring of Gyges for their own personal benefit?

As a physicist working in the field of quantum information and quantum computation I believe that these are some questions that need to be addressed not only in the scientific community, but along with private companies and nation governments. I know that with so much at stake it is a sensitive subject, but fortunately quantum computers are still in their infancy and we still have time to raise the issue.

If you see what I see, if you feel as I feel, and if you would seek as I seek, then I ask you to stand beside me.

[1] Plato, the Republic, Book II, 358d—361d.

[2] Peter W. Shor, *Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer*

[3] NSA Says It “Must Act Now” Against the Quantum Computing Threat

[4] NSA memorandum

[5] Australia: Govt Prepares for Quantum Computing Threat to Encryption

[6] Prepare for Threat of Quantum Computing to Encrypted Data, Canadian Conference Told

[7] Here, There and Everywhere, Quantum technology is beginning to come into its own

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

Why we should be talking about the ethics of quantum computation by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Based on a work at https://entangledphysics.com/.

Nothing that is done in one system can instantaneously affect the density matrix of another isolated system, although it can affect the state vector.

Nobel Laureate Steven Weinberg, [1]

[1] “Quantum mechanics without state vectors”, Steven Weinberg, Phys. Rev. A **90**, 042102 (2014).

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One of the most important consequences of the existence of *entanglement* in quantum mechanics is related to the *quantity of information* necessary to describe the state of a quantum system. In a classical system, the number of parameters needed to characterize its state is equal to the sum of the degrees of freedom of each of its components. However, for a quantum system, the number of parameters needed to describe its state grows *exponentially* with the number of constituents. And this makes the task of simulating a quantum system a very difficult one even for systems with few constituents, the computational resources needed for the simulation quickly become unreachable using computational techniques based on classical computers.

In 1982 Nobel laureate Richard Feynman conjectured that these limitations could be overcome using computers based on quantum systems. Based on this conjecture, physicists started exploiting the potential of quantum mechanics to develop new quantum computers (**universal quantum simulators**) capable of simulating quantum systems.

Nowadays there are several quantum systems than can be used as hardware for quantum simulators: trapped ions, cold atoms in optical lattices, liquid and solid-state NMR, photons, quantum dots, superconducting circuits, and NV centres. [1]

Go to the Dictionary of Quantum Information and Quantum Computation.

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

The More You Know: Quantum Simulator by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Although my first thought was to use a generic drawing software the results were not at all satisfying. Thus I proceeded to take the next logical step: an online search for a Quantum Circuit drawing tool. In this article I review two particularly interesting tools I found.

I first came across this website by programmer Davy Wybiral, which is an *online based* **Quantum Circuit Simulator**. It’s interface is very clean and the circuit can be easily and intuitively created by simply selecting the quantum gates and placing them in the circuit. This website produces results such as the following

Additionally, as its name indicates, this website can also perform a simulation of the quantum circuit. For instance, the previous circuit generates the Greenberger-Horne-Zeilinger (GHZ) state

, (1)

and its simulation yields

0.70710678+0.00000000i|000>50.0000%0.70710678 + 0.00000000i|111>50.0000%,

which coincides with Eq. (1).

Although the circuits created with this website are quite nice, there are some operations that are not available and thus cannot be drawn (like measurements). I recommend this website for anyone who needs to quickly draw a Quantum Circuit and who does not know how to use .

A deeper search led me to many downloadable softwares, but in particular it led me to Michael Nielsen’s blog where he recommends this LaTeX package for drawing Quantum Circuits. As cited in their website this tool is

[…] a simple text-format language for describing acyclic quantum circuits composed from single qubit, multiply controlled single-qubit gates, multiple-qubit, and multiple-qubit controlled multiple-qubit gates.

Figures of quantum circuits in the book “Quantum Computation and Quantum Information,” by Nielsen and Chuang, were produced using an earlier version of this package.

Needless to say, after a couple of minutes reviewing their *17 examples*, I was in love with this tool. Once you get the hang of it it’s pretty simple to use and the results are just amazing!

I would *strongly recommend* MIT LaTeX package to anyone who knows how to use and who needs to draw Quantum Circuits on a regular basis.

Although I’m sure there are software out there available for people without programming or experiencie I didn’t try them all out. If any of you has any recommendations I’d be glad to review them!

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

Tools for Drawing Quantum Circuits by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

The data in a Quantum Circuit is usually encoded in a -qubit state (other Quantum Circuit models can use -level quantum systems instead of these -level systems) which is set to a known vale, i.e. . Thus, for every qubit in the circuit a horizontal rail, a *wire*, is drawn to represent its evolution, time flows from left to right.

Additionally, a Quantum Circuit is built from elementary quantum gates (unitary operations) acting on a fixed number of qubits. Since Quantum Circuits are read from left to right, leftmost gates act first.

Then, a description of the Quantum System consists of

**Input**: A controlled input -qubit state.**Evolution**: Prescribed sequence of Unitary Single-qubit and Two-qubit logic quantum gates which give the evolution of the initial quantum state.**Output**: Readout (measurement) of the state of the qubits.

Go to the Dictionary of Quantum Information and Quantum Computation.

Quantum Circuits where drawn using this LaTeX package.

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

The More You Know: Quantum Circuit por Marco Vinicio Sebastian Cerezo de la Roca se distribuye bajo una Licencia Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional.

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In the *computational basis* the four maximally entangled Bell states are expressed as

,

.

To understand the correlations present in these states consider the Bell state . If a measurement is performed on the first qubit the result may be 0 or 1, after which the two qubit state collapses to or respectively. Then, a measurement on the second qubit *must* result in 0 or 1, yielding the same result as the measurement on the first one. That is, the outcomes of the measurements are *correlated*.

John Bell proved that the correlations present in EPR pairs* cannot be accounted for by any classical theory*. See this article for more.

Go to the Dictionary of Quantum Information and Quantum Computation.

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

The More You Know: Bell States/EPR Pairs por Marco Vinicio Sebastian Cerezo de la Roca se distribuye bajo una Licencia Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional.

,

where and are complex number such that . Since an overall phase has no observable effects, or can be chosen as real numbers and can written as

,

where and . The parameters and define a point on the sphere called the Bloch Sphere (Fig. 1).

The points in the *surface* of the sphere represent *pure* states of the qubit, with as the North pole and as the South pole, and the equator states of the form . Additionally any *mixed state* of a qubit can be represented as a point *inside* the sphere, with maximally mixed states at the center of the Bloch Sphere.

This representation is useful in Quantum Computation to visualize the effect of single qubit gates.

Go to the Dictionary of Quantum Information and Quantum Computation.

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

The More You Know: Bloch Sphere por Marco Vinicio Sebastian Cerezo de la Roca se distribuye bajo una Licencia Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional.

Image credits here.

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