# How to program a quantum computer

## Battleships with partial NOT gates

Quantum computers are a weird new kind of computer. They burn through unthinkable numbers of parallel universes to run programs faster, and use principles that confounded even Einstein. They are the magical miracle boxes that will leave your stupid boxes of transistors in the dust.

That’s what popular science articles will probably tell you, anyway. They doubtless succeed in making this new technology sound exciting. But they can also make quantum computation seem like an arcane art, entrusted to only the cleverest of scientists. Certainly not something for the wider programming public to concern themselves with!

I don’t think this is true at all. And with companies like IBM and Google actually making quantum devices, the time to start playing with quantum programming is now.

You don’t need to do anything fancy at the beginning. You can start your journey into quantum programming at the same place as many of us start our journey into normal programming: by making games.

Don’t worry, having your own quantum computer will not be required. Simple quantum programs can easily be simulated on a normal computer. We can also borrow some time on a real quantum device, thanks to IBM’s Quantum Experience.

This is a first in a series of articles in which I describe some simple programs made using IBM’s quantum API. Each will be a self-contained version of Battleships, with simple quantum processes used to implement the basic game mechanic.

These are among the first games made for a quantum computer. I’ll show you the gory details of the code behind the game, so that hopefully you can start experimenting with quantum programming too.

What is a quantum computer?

Before we start on the program, we need some background information. I won’t tell you everything about quantum computing. We’ll just get enough to understand what goes on in our game.

A normal computer is based on bits: variables which have just two possible values. We often call them 0 and 1, though in the context of Boolean algebra we might call them True and False. It doesn’t matter what we call them. The important point is that there are just two states.

With bits we can do simple Boolean operations, like NOT, AND and OR. These are the basic building blocks of computation. With these we can do anything. Any variable more complex than a bit (like an int or float) is just collection of many bits. Any operation more complex than an AND or NOT is actually just many of them stuck together. At its most basic level, this is what a normal computer is.

Quantum computers are based on quantum bits, or qubits. These also have two possible values which we can call 0 and 1. But the laws of quantum mechanics also allow other possibilities, which we call superposition states.

In some sense, the superposition states are values that exist part way between the extremes of 0 and 1. We can picture a qubit as a sphere, with 0 and 1 sitting on opposing poles. The superposition states are all the other possible points on the surface.

As well as 0 and 1, this image also flags up a couple of important superposition states. One is called u3(0.5*pi,0,0) 0. Admittedly it’s not a very catchy name, but we’ll look at what it actually means during this article. The other is h 0, and we’ll learn more about that next time.

This way of thinking about qubits makes them seem a bit like continuous variables. We can represent any point on the surface of a sphere (like the point ψ in the image) using polar coordinates with a just couple of angles (θ and φ). So you’d be forgiven for thinking that a qubit is just a pair of floats. In a sense, they are. But in another more accurate sense, they are not.

An important difference is that we can never extract more than a binary piece of information from a qubit. The laws of physics themselves prevent us from finding out what exactly they are doing. We can’t ask a qubit for the exact details of its superposition state. We can only ever force it to choose between two opposite points on the sphere (like 0 and 1). If it is anything other than these two states, it will just have to randomly decide one or the other.

So a qubit has some properties of a continuous variable, and some properties of a discrete one. In truth it is neither. It is a quantum variable.

What will we do with the qubits?

The Battleships game we are making today is based on a Japanese variant. In this, all ships take up only a single square, but some take more hits to sink than others. We’ll have one ship that can be sunk by a single bomb, one that needs two and one that needs three.

To simulate this on a quantum computer, we can use a qubit for each ship. The state 0 can be identified with a fully intact ship, and 1 with one that has been destroyed. The superposition states will then correspond to a ship’s journey towards destruction.

A ship destroyed by a single hit will be pretty easy to simulate. We can initialize it in state 0, and then apply a NOT when it gets hit. When we find it in state 1, we’ll know it has been destroyed.

Let’s look at how to implement just this simple process on a quantum computer. We won’t worry about any other ships, or inputs and outputs at the moment.

A basic assembly language for quantum computers is called QASM. Here’s a QASM script that initializes a ship, bombs it, and then looks at whether it is still afloat.

OPENQASM 2.0;include "qelib1.inc";
qreg q[1]; \\ Initialize a register with a single qubitcreg c[1]; \\ Initialize a register with a normal bit        u3(pi,0,0) q[0]; \\ apply a NOT to the qubitmeasure q[0] -> c[0]; \\ measure the qubit

The first two lines are always there at the top of any proper QASM file. After this we then define all the bits we’ll need: both quantum and normal. In this example we have defined a single qubit in a register called q. We refer to this qubit in code as q[0]. Since outputs have to be in nice, human-readable normal information, we also define a single normal bit in a register called c.

The qubit q[0] is automatically initialized in the state 0. Since that’s the state we want to start with, no further preparation is required.

Next we want to take this q[0] = 0, a fully intact ship, and perform a NOT. With a normal bit and a normal programming language, we might implement this as q[0] = !q[0] (for C++) or q[0] = not q[0] (for Python). In QASM it can be done in multiple ways. The simplest is to use operations called x and y. Here are some examples of how to use them, with the equivalent C++ and Python lines for comparison.

q[0] = !q[0]; \\ A C++ style NOT
q[0] = not q[0] # A Python style NOT
x q[0]; \\ A QASM style NOT
y q[0]; \\ Another QASM style NOT

There are some differences between x and y that we’ll have to deal with one day. But not today.

You’ll probably have noticed that neither x nor y appear in the earlier QASM file. Instead we have u3(pi,0,0). This is another way to implement a NOT.

u3(pi,0,0) q[0]; \\ Another QASM style NOT

It’s completely equivalent to an x. But u3 is a more complex operation in general. It has three arguments, and by changing them we can do other things.

The first argument is an angle expressed in radians. It is the angle by which we are going to turn the sphere of our qubit around. The angle pi corresponds to 180°, and so means we turn the sphere completely upside down. 0 moves to 1 and 1 moves to 0, which is why this operation acts as a NOT. To do half a NOT we could simply use half this angle: u3(0.5*pi,0,0).

So now we have yet another way to perform a NOT on our qubit: We could do half a NOT twice.

\\ Another QASM style NOTu3(0.5*pi,0,0) q[0];u3(0.5*pi,0,0) q[0];

We could also do a third of a not thrice, or a quarter of a NOT following by half a NOT and then two eighths of a NOT. You get the idea.

The last line of our QASM file is

measure q[0] -> c[0];

In this we measure the qubit. We tell q[0] that it has to decide what to be: 0 or 1. The result is then stored in c[0], ready to be looked at by our non-quantum brains, or processed by our non-quantum computers. The value of c[0] is then the output of this computation.

If we did nothing between initialization and measurement…

qreg q[1];creg c[1];measure q[0] -> c[0];

…the result should always be c[0] = 0.

If we did a NOT…

qreg q[1];creg c[1];        u3(pi,0,0) q[0];measure q[0] -> c[0];

..the result should always be c[0]=1.

If we did just half a NOT…

qreg q[1];creg c[1];        u3(0.5*pi,0,0) q[0];measure q[0] -> c[0];

q[0] will be halfway between 0 and 1 when it is measured. In fact it will be the superposition state we called u3(0.5*pi,0,0) 0 earlier. Measuring whether this is 0 or 1 will force it to randomly choose one or the other, with equal probabilities.

This probabilistic nature will crop up a fair bit in quantum calculations. Additional randomness is also present in current devices due to noise. Though this is at a usuably low level, we still have to keep its existence in mind. So we will sometimes get a 1 when we should get a 0, and vice-versa.

For these reasons, it is usual to run a quantum program many times. This process then returns a list of all the outputs that were generated during these many samples, and the number of times each occurred.

#### Making a game

Now let’s use these many paths toward a NOT to make a game on a quantum computer.

First, I’ll let you into a secret. Quantum computers will always be hybrid devices, partly quantum and partly normal. The latter is required to handle inputs and outputs, in order to interface with the lumbering apes who want to use the device.

This hybrid nature will have an effect on how we program. Using QASM is great for the quantum part, but it would be nice to use something more familiar to do the rest. Like Python, for example.

This is the approach taken by the current version of IBM’s quantum API. And it’s the approach I use in these tutorials. Other methods are available, like Project Q.

It’s important to note that everything is still evolving. And all of us making quantum programs in these early days will inevitably guide that evolution.

To run the program, you’ll need Jupyter installed, as well as the dependencies of the SDK. You can find out more here.

Now let’s look at the code!

Firstly, everything we need to run code on the IBM’s Quantum Experience needs to be imported.

import syssys.path.append("../../")from qiskit import QuantumProgramimport Qconfig

This references the Qconfig file, which you’ll need an account with IBM to set up. Don’t worry, it’s free! See the guide on IBM’s SDK for the specifics.

Next follows a couple of standard imports and some information about the game printed to screen, which you should probably be fine with. But there is also a question.

d = input("Do you want to play on the real device? (y/n)\n")if (d=="y"):    device = 'ibmqx2'else:    device = 'local_qasm_simulator'# note that device should be 'ibmqx_qasm_simulator', 'ibmqx2' or 'local_qasm_simulator'

The player is asked what device should be used to run the quantum part of the circuit. There are three main options. One is to simulate the quantum stuff on your own computer (local_qasm_simulator). You could also ask IBM’s computers to do the simulating for you (ibmqx_qasm_simulator). But you could also use the real 5 qubit quantum device: ibmqx2. We ask the player whether it wants to use this, or to just simulate it locally. The process is never complicated enough that we’ll need IBM to do the simulating for us.

We then set another important variable: the number of times each job will be run (to collect the statistics).

# while we are at it, let's set the number of shotsshots = 1024

There is no magical reason for 1024. You can change it if you want.

Next we come to setting up the boards, with each player choosing where to put three ships. Five possible positions are available, corresponding to the five qubits on IBM’s ibmqx2 chip. I visualize the grid like this

4       0|\     /|| \   / ||  \ /  ||   2   ||  / \  || /   \ ||/     \|3       1

The numbers are the names I use for each position. They are also the names IBM uses, so 2 is the position of qubit q[2], for example.

The choices made by the players are stored in shipPos. This has an entry for each of the two players (player=0 for the first player, player=1 for the second), and each of the three ships they place. The entry shipPos[player][ship] holds the position (0, 1, 2, 3 or 4) for the ship numbered ship belonging to player.

Getting the choices from the players is basically done by

position = getpass.getpass(“Player “ + str(player+1) + “, choose a position for ship “ + str(ship+1) + “ (0, 1, 2, 3 or 4)\n” )
shipPos[player][ship] = position

This prints “Player player, choose a position for ship ship+1 (0, 1, 2, 3 or 4)” and asks for an input. By using getpass, the input won’t be visible on screen for the other player to see. Note that the first ship is called ship=0 in code, but referred to as ‘ship 1' when talking to the player, since most humans index from 1 (fools!).

These two lines are embedded in a couple of loops over the players and ships, to extract all the required values. They also aren’t actually next to each other in the real code. There is some stuff in between to ensure that the player gave a valid input. Your way of implementing this would probably be better than mine, so I won’t bore you with the details.

Now it’s time for the main game loop. In this we will ask both players where on their opponent’s grid they’d like to bomb. This is then added into bomb[player][position], which counts how many times player has bombed position over the course of the game.

position = input("Choose a position to bomb (0, 1, 2, 3 or 4)\n")
bomb[player][position] = bomb[player][position] + 1

We now have enough information to run the process on the quantum chip. We know which qubits are supposed to be ships and where to bomb. So now it is time to create the corresponding QASM file.

OPENQASM 2.0;include "qelib1.inc";qreg q[5];creg c[5];

This declares the quantum and normal bits we’ll need. We declare five of each, even if we won’t use them all. Using five qubits is a reflection of the fact that we know 5 qubits actually exist on the chip, and it allows us to identify each ship with an actual physical qubit. Using five normal bits is something we can do to avoid a bug that exists on IBM’s end at the moment.

Instead of writing the QASM file directly, we’ll get Python to write it for us. So instead of the above preamble, we define a dictionary which holds the same specifications.

Q_SPECS = {        "circuits": [{            "name": "gridScript",            "quantum_registers": [{                "name": "q",                "size": 5            }],            "classical_registers": [{                "name": "c",                "size": 5            }]}],}

Here the file has been called gridScript, because it will calculate the effects of the bombs on the grid of a player.

Once we have the specs set up, we can create the circuit and start to manipulate it.

# create the program with these specsQ_program = QuantumProgram(specs=Q_SPECS) # get the circuit by namegridScript = Q_program.get_circuit("gridScript")

Here “gridScript” is what we called the circuit in the specs, and gridScript is what we’ll refer to it as when adding operations to it. We could have used different names for these two things in general. But we didn’t.

In order to add in operations on the quantum and classical bits, we’ll need to define how we refer to them in code.

# get the quantum register by nameq = Q_program.get_quantum_registers("q")# get the classical register by namec = Q_program.get_classical_registers("c")

Now suppose we want to apply a certain fraction of a NOT to a certain qubit. This means we’d want to add the following line to gridScript,

u3( frac * pi,0,0) q[ position ];

but with frac and position replaced by actual numbers. We do this with the following line in Python.

gridScript.u3(frac * math.pi, 0.0, 0.0, q[position])

Using this we can add all the bombs that the opposing player is firing at this grid. For the grid of player we loop over all three positions were there is a ship. We then add a line to gridScript for every bomb that has been sent here.

I mentioned earlier that the three ships in this game will all have different strengths. The first one placed by the player will be sunk by a single bomb, but the second one will need two to be destroyed and the third will need three.

In terms of qubits, this means that a bomb that hits the first ship applies a whole NOT. A bomb that hits the second ship applies, half a NOT, and one on the third ship applies a third of a NOT. The fraction frac is then determined by what ship is getting hit. For this we simply use frac = 1/(ship+1).

Once the bombing is over, it’s time to see what happened. For that we need to add the measurement commands. Each qubit in the register is measured, and the results copied to a corresponding normal bit. This is done with

for qubit in range(5):            gridScript.measure(q[qubit], c[qubit])

This creates the QASM lines

measure q[0] -> c[0];measure q[1] -> c[1];measure q[2] -> c[2];measure q[3] -> c[3];measure q[4] -> c[4];

Now we have a complete QASM file stored as gridScript. It’s time to run it. This is done using

# set the APIToken and API urlQ_program.set_api(Qconfig.APItoken, Qconfig.config["url"])# compile and run the QASMQ_program.execute(["gridScript"], device, shots, wait=2, timeout=60)

Here device and shots were defined earlier when we asked the player which device to run on. You’ll be updated on the progress of your runs every wait seconds, and they’ll give up after timeout seconds.

Here I assume you don’t want to wait more than a minute for results. But if you actually want to play on the quantum device, you might have to extend this. Quantum computers might be fast, but there will probably be some jobs to run before yours.

Once the jobs have run, we can extract the data.

grid[player] = Q_program.get_counts("gridScript")

Here I copy the results over to grid, with the results for the bombing of player’s ships in grid[player].

The results are stored in grid[player] as a dictionary. The keys are bit strings, and so look something like 110. The rightmost bit is c[0], and so c[0]=0 in this example bit string. By processing the results we can work out how many runs got the each result (0 or 1) for each of the normal bits.

We interpret the fraction of times a qubit is measured to be 1 as the percentage damage for our ship. We can’t expect 100% damage due to the effects of noise (or weather, if you want an in-universe explanation) so we count damage greater than 95% as meaning the ship has sunk.

Let’s check out some example outputs from the quantum process. Suppose one player put their third ship at position 2. The other player then bombs it. The results will look something like.

{'00000': 734, '00100': 290}

Here 290 of the 1024 runs found a 1 at position 2, showing that this ship did indeed suffer some damage.

Another example: one player put their first ship at position 1, and their second at 2. The other player then bombs position 1 in the first round and position 2 in the second. The results would be

{'00110': 532, '00010': 492}

Here all results have a 1 on the second slot to the right (the one for position 1). That ship was destroyed by a single bomb. The one at position 2 was only half damaged, though, so half the results have a 1 for that.

These example results were nice and clean, so we can tell that they were done on the simulator. On the real device you can expect to find a few 1 results for ships that weren’t even bombed, and for qubits that aren’t ships.

If there was an error when executing the QASM file on the quantum chip (or simulator) the output will include

{'status': 'Error'}

In this case we don’t want to try and process the results. So we set up a conditional statement to work out if this happened.

if ( ( 'Error' in grid[0].values() ) or ( 'Error' in grid[1].values() ) ):
   print("\nThe process timed out. Try this round again.\n")

It is in the else of this statement that all the processing of results is done, as well as telling the players what’s happened with their ships.

The processing of bit strings is not something unique to quantum computing, so I won’t presume to tell you how to do it. If you are interested in the method I hacked together you’ll find it in the code.

Once the damage percentages are calculated, they are presented to the players. For example, if only the ship at position 2 has been hit, and it has 50% damage, this would be shown as.

 ?       ?  |\     /| | \   / | |  \ /  | |  50%  | |  / \  | | /   \ | |/     \| ?       ?

The game continues to ask for bombs and running the scenario until one player has lost all their ships. At that point the other player wins.

You might notice that bombing the third ship of a player gives a damage of around 25%. Since this ship is a third of the way to destruction, you might have expected 33%. The reason is trigonometry. There’s no need to dwell on it too much at this point.

It is important to note that the whole scenario is rerun every time. If the amount of time between turns might only be about minute, but that’s almost an eternity for qubits. Even at a temperature of 0.02 Kelvin, the information would have burned away long before the next turn. Also, our need to extract information will also disturb all the lovely quantumness in the system. For these reasons, any process that needs human interaction throughout will need the quantum parts to be rerun from scratch for each new input.

So that was it. Battleships running on a quantum computer. Not the fanciest use of a quantum computer, or the fanciest version of Battleships. But for me, combining the two is half the fun!

The full source code can be accessed below. Note: This is based on a slightly old version of the SDK. Minor changes to the article and a new link to the source code will come soon.

And here’s a video of the game in action.

The story continues with another version of Battleships in part 2.