tutorial/001_qubit.ipynb

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    "$$\\renewcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$$\n",
    "$$\\renewcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$$\n",
    "\n",
    "# Quantum States and Quantum Logic Gates Basics\n",
    "Quantum computation is usually carried out by initializing a quantum state consisting of a number of two-level quantum mechanical systems called qubits, operating several quantum logic gates on it, and performing a measurement on the final quantum state. \n",
    "\n",
    "The way the state is implemented as a physical system depends on the architecture of the individual quantum computer and the same goes for quantum gates and measurements.\n",
    "\n",
    "Let us see how quantum states are represented by qubits and quantum logic gates act on them more in detail in the following."
   ]
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   "source": [
    "## Qubit and quantum states\n",
    "A qubit is a quantum system which has two levels, usually denoted symbolically as $\\ket{0}$ and $\\ket{1}$. Since it is a quantum system, a general state $\\ket{\\psi}$ for one qubit is represented by a linear combination of these two basis states with complex coefficients $\\alpha$ and $\\beta$.\n",
    "\n",
    "$$\n",
    "\\ket{\\psi} = \\alpha \\ket{0} + \\beta \\ket{1}\n",
    "$$\n",
    "\n",
    "Since state vectors are normalized, the coefficients satisfy $|\\alpha|^2 + |\\beta|^2 = 1$. Also, the overall phase factor of $\\ket{\\psi}$ is physically irrelevant. Thus we can always choose the phase of $\\alpha$ to be real.\n",
    "\n",
    "It is convinient to express $\\alpha$ and $\\beta$ with real angle parameters $\\theta$ and $\\phi$;\n",
    "$$\n",
    "\\alpha = \\cos(\\frac{\\theta}{2}),~ \\beta = e^{i\\phi}\\sin(\\frac{\\theta}{2})\n",
    "$$\n",
    "\n",
    "This means one qubit can be represented as a point on a sphere. This sphere is called the Bloch sphere. A quantum state represented by qubit is not constrained to only 0 and 1 but rather can take any point on this sphere.\n",
    "\n",
    "Somewhat abstract state vector $\\ket{\\psi}$ can be expressed as a more concrete row vector with complex components once we fix the basis of qubit vector space to $\\ket{0}$ and $\\ket{1}$.\n",
    "\n",
    "$$\n",
    "\\ket{0} = \\begin{bmatrix}1\\\\0\\end{bmatrix}, \\ket{1} = \\begin{bmatrix}0\\\\1\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "This way, $\\ket{\\psi}$ is also represented by a complex row vector:\n",
    "\n",
    "$$\n",
    "\\ket{\\psi} = \\alpha \\ket{0} + \\beta \\ket{1} = \\alpha \\begin{bmatrix}1\\\\0\\end{bmatrix} + \\beta \\begin{bmatrix}0\\\\1\\end{bmatrix} = \\begin{bmatrix}\\alpha \\\\ \\beta \\end{bmatrix}.\n",
    "$$\n",
    "\n",
    "This choice of basis is the most commonly used one and called computational basis."
   ]
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    "## Initial state\n",
    "The simplest choice for an initial state would be:\n",
    "\n",
    "$$\n",
    "\\ket{0} = \\begin{bmatrix}1\\\\0\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "This is the default choice for the initial state vector in the Blueqat toolkit and implicitly used unless it is explicitly initialized to some other state.\n"
   ]
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    "\n",
    "## Quantum gates as rotations\n",
    "\n",
    "The quantum state is modified by the quantum circuit set up for a specific purpose before it is measured to give the final outcome.\n",
    "\n",
    "The component blocks of a quantum circuit are quantum gates, often called unitary gates since methematically they cause unitary transformations on quantum states.\n",
    "\n",
    "Unitary transformation is a generaliztion of rotation in complex vector space. Unitary transformation for one qubit is comprised of rotations along X, Y, and Z axis of the Bloch sphere, except for physically irrelevant overall phase factor.\n",
    "\n",
    "These three basic rotation operations are generated by the X, Y, and Z gates also known as Pauli gates.\n",
    "\n",
    "$$\n",
    "X=\n",
    "\\begin{bmatrix}\n",
    "0&1 \\\\\n",
    "1&0\n",
    "\\end{bmatrix},\n",
    "Y=\n",
    "\\begin{bmatrix}\n",
    "0&-i \\\\\n",
    "i&0\n",
    "\\end{bmatrix},\n",
    "Z=\n",
    "\\begin{bmatrix}\n",
    "1&0 \\\\\n",
    "0&-1\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Rotations around these axes are represented by the exponentials of them:\n",
    "\n",
    "$$\n",
    "R_x(\\theta) = e^{-\\frac{i\\theta}{2}X},~\n",
    "R_y(\\theta) = e^{-\\frac{i\\theta}{2}Y},~\n",
    "R_z(\\theta) = e^{-\\frac{i\\theta}{2}Z}\n",
    "$$\n",
    "\n",
    "There are some more basic quantum gates used frequently in quantum computation. Let us see how they work along with the rotation gates in the following examples using the Blueqat toolkit. For more through introduction refer to [3]."
   ]
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    "# Blueqat quickstart\n",
    "## Installation\n",
    "The following is a quick tour of the basic functionalities of the Blueqat quantum computation toolkit. First, if you haven't yet done so, install the blueqat package as you do for any python package:"
   ]
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   "source": [
    "!pip install blueqat"
   ]
  },
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    "## Step1: Prepare basic circuit\n",
    "To perform a calculation on a quantum computer, you must create an instance for the quantum circuit. Let's import the main component of the Blueqat toolkit and instantiate one. The quantum state is implicitly initialized to $\\ket{0}$."
   ]
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Circuit(0)."
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from blueqat import Circuit\n",
    "\n",
    "#initialization of circuit\n",
    "Circuit()"
   ]
  },
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    "## Step2: Operate quantum logic gate on the circuit\n",
    "Next, we place some quantum logic gates on the circuit.\n",
    "You can put quantum logic gates by chaining methods starting from the initialized circuit. The .x[0] in the example means that we apply the X gate on the 0th qubit."
   ]
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    {
     "data": {
      "text/plain": [
       "Circuit(1).x[0]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().x[0]"
   ]
  },
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    "## Step 3: Measurement\n",
    "To get the result of this circuit, you have to perform a measurement on it. Let's attach a measurement gate as .m[:]. ':' indicates that we apply the measurement gate to all of the qubits. The colon notation is borrowed from numpy array one. In this small example .m[0] would have given the same result. In the Blueqat toolkit measurements are always done in Z basis, i.e. always result in 0 and 1."
   ]
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  {
   "cell_type": "code",
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   "metadata": {},
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   "source": [
    "Circuit().x[0].m[:]"
   ]
  },
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   "source": [
    "## Step 4: Running the circuit\n",
    "Finally, append a .run() method to actually perform the quantum computation process designated by this circuit. The result of quantum measurement is a random variable and its value obtained varies in each run, so usually you need to run the circuit multiple times to accumulate the results to see the average. You can specify the number of runs with the \"shots\" argument in the run() method. Currently, the result is returned as a python Counter object by default, showing the distribution of the results."
   ]
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'1': 1})"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
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   "source": [
    "Circuit().x[0].m[:].run(shots=1)"
   ]
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   "source": [
    "You should get the result '1' once. Usually, the circuit starts from 0 and in our case applying the X gate flips the qubit to 1. That's why you get 1 as a final result.\n",
    "\n",
    "You can also show the circuit diagrammatically by giving 'backend=\"draw\" ' option to the run() method."
   ]
  },
  {
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   "execution_count": 3,
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      "text/plain": [
       "<Figure size 1800x120 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit().x[0].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## basic 2 qubits operations\n",
    "Here we learn about basic 2 qubits operations. 2 qubits operations let you perform more complex computations and make it possible to implement various algorithms by creating correlations between qubits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating and running the circuit\n",
    "The preparation of the circuit is the same as the 1-qubit example. You don't need to set the number of qubits explicitly when you instantiate the circuit. The Blueqat toolkit checks the structure of the circuit and decides how many qubits are required to implement the calculation.\n",
    "In this example, we use a CX gate working on qubits 0 and 1, so this is a 2-qubit circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'00': 1})"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().cx[0,1].m[:].run(shots=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x180 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit().cx[0,1].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get the result of '00' once. 00 means that both of the qubits are 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CX gate\n",
    "The CX quantum logic gate is called control X gate, control NOT gate, or CNOT gate for short. This gate works on two qubits.\n",
    "One is called \"control qubit\" and the other is \"target qubit\". Only when the control qubit is '1' does the target qubit flip. If the controlled gate is '0' the target qubit is unchanged.\n",
    "In this example, we have both qubits initialized to 0 so the target qubit stays in state 0 after applying the CX gate. The qubit designated by the first index serves as the control qubit and the second index designates the target qubit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CX gate continued\n",
    "\n",
    "Let's see what happens if the control qubit is 1. Applying an X gate flips it and after that, we apply a CX gate on qubits 0 and 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'11': 1})"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().x[0].cx[0,1].m[:].run(shots=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x180 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit().x[0].cx[0,1].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have the result '11'. First, the circuit flips the 0th (control) qubit to 1 with the .x[0] gate and then the .cx[0,1] gate flips the 1st (target) qubit from 0 to 1, resulting in 1 for both qubits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Advanced topics\n",
    "## initializing circuit with a definite number of qubits\n",
    "Usually, the circuit automatically determines the number of qubits used in itself, but we can also specify the number of qubits at the creation time.\n",
    "\n",
    "To do so, put the number of qubits as an argument in the Circuit() method. In the following example 3 qubits are prepared and only two of them are used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'110': 1})"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit(3).x[0].cx[0,1].m[:].run(shots=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x240 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit(3).x[0].cx[0,1].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Superposition\n",
    "Here we see a basic example of quantum superposition. Superposition is mathematically just a linear combination of basis state vectors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Circuit\n",
    "The superposition is quite easy to implement. You can just apply a Hadamard gate (H gate) to the qubit which is implicitly initialized to state $\\ket{0}$.\n",
    "$$ H = \n",
    "\\frac{1}{\\sqrt{2}}\n",
    "\\begin{bmatrix}\n",
    "1 & 1\\\\\n",
    "1 & -1\n",
    "\\end{bmatrix}\n",
    "$$\n",
    " The H gate transforms the state to $(\\ket{0} + \\ket{1})/\\sqrt{2}$. Measuring this state to see if the result is 0 or 1 would give each outcome with equal probability of 0.5.\n",
    "To see the result we run the circuit 100 times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'0': 47, '1': 53})"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().h[0].m[:].run(shots=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x120 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit().h[0].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get results 0 and 1 almost half each other. The actual number can fluctuate slightly from exact 50:50 due to randomness, but it is in accordance with probability for 0 and 1 both being $(\\frac{1}{\\sqrt{2}})^2 = 0.5$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Entanglement\n",
    "Entanglement is the source of correlation between two or more quantum systems. Consider the following two-qubit quantum state for example.\n",
    "\n",
    "$$\n",
    "\\ket{\\psi} = (\\ket{00} + \\ket{11})/\\sqrt{2}\n",
    "$$\n",
    "\n",
    "In the above, $\\ket{00}$ and $\\ket{11}$ are shorthand for $\\ket{0}\\ket{0}$ and $\\ket{1}\\ket{1}$ (or $\\ket{0}\\otimes\\ket{0}$ and $\\ket{1}\\otimes\\ket{1}$ even more formally) respectively.\n",
    "\n",
    "Suppose the system is in the state $\\ket{\\psi}$ and we measure the first qubit tp obtain the result 0. Then we can tell the second qubit is also in state 0 for sure. Similarly, if we get 1 for the first qubit measurement, the second qubit is also certainly in state 1, i.e. these two qubits are correlated -- in other words, entangled.\n",
    "\n",
    "As a quantum state is passed through a complex quantum circuit that includes many gates the number of non-zero terms in computational basis contained in the state can get quite large, typically $O(2^n)$ where $n$ is the number of the qubits.\n",
    "\n",
    "Measuring a qubit and obtaining a definite result for the qubit picks up only the terms that correspond to the obtained value of the measured qubit. That way we can pull out only results that suit the purpose of the calculation to be carried out.\n",
    "\n",
    "The short example below shows a very simple case of this phenomenon."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Circuit\n",
    "\n",
    "Our circuit is a very simple one. We just apply the CX gate after the H gate. This circuit creates an entangled state $\\ket{\\psi}$ mentioned above.\n",
    "$$\n",
    "\\ket{\\psi} = (\\ket{00} + \\ket{11})/\\sqrt{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'11': 53, '00': 47})"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().h[0].cx[0,1].m[:].run(shots=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x180 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Circuit().h[0].cx[0,1].m[:].run(backend=\"draw\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now you get about half times of '00' and '11' respectively. If it were just a product of superposition states of each qubit\n",
    "\n",
    "$$\n",
    "\\ket{\\psi^\\prime} = (\\ket{0} + \\ket{1})(\\ket{0} + \\ket{1})/2\\\\\n",
    "= (\\ket{00} + \\ket{01} + \\ket{10} + \\ket{11})/2\n",
    "$$\n",
    "\n",
    "for example, we would have obtained a result somewhat close to {'00':25, '01':25, '10':25, '11':25} instead."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The GHZ state: entanglement of 3 qubits\n",
    "Greenberger–Horne–Zeilinger state is an entangled state comprised of three qubits as follows:\n",
    "$$\n",
    "\\ket{GHZ} = (\\ket{000} + \\ket{111})/\\sqrt{2}\n",
    "$$\n",
    "This state can be created by the following circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'111': 48, '000': 52})"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Circuit().h[0].cx[0,1].cx[1,2].m[:].run(shots=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have about half '000' and '111' for another half."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantum algorithm\n",
    "\n",
    "### Overview\n",
    "\n",
    "An algorithm generally refers to a procedure that a computer performs in order to solve a certain problem.\n",
    "\n",
    "A quantum algorithm is an algorithm that is designed to be run on a quantum computer.  \n",
    "It is often described by mathematical expression using bracket symbols, but eventually, it is reduced to a quantum circuit using concrete quantum gates.  \n",
    "\n",
    "The aim of quantum algorithms is to perform calculations that are faster than classical computers, or that are virtually impossible with classical computers due to physical constraints (e.g., memory capacity) using quantum computers.  \n",
    "This is achieved by utilizing the superposition and entanglement properties of qubits.  \n",
    "Unlike (classical) qubits, $n$ qubits can hold $2^n$ superposition states simultaneously.  \n",
    "\n",
    "At first glance, it is tempting to think that $2^n$ of calculations can be performed in parallel, but unfortunately, it is not so simple.  \n",
    "This is because, in a quantum computer, only one of the $2^n$ states is probabilistically output by the \"measurement\" process.  \n",
    "In other words, a quantum computer can hold a huge amount of information internally, but the information it can output is very limited.  \n",
    "Therefore, quantum algorithms need to be designed to be good at obtaining the desired solution with a small number of outputs.  \n",
    "\n",
    "Currently, there are only a very limited number of calculations where quantum algorithms have been shown to be faster than classical algorithms.  \n",
    "Included in that limited number of examples are the famous Shor's and Grover's algorithms.  \n",
    "In the process of learning these algorithms, I think you will be able to grasp the kind of calculations in which a quantum computer is likely to have an advantage over a classical computer.　　\n",
    "The discovery of new computations that will make quantum algorithms faster will be the driving force behind the use of quantum computers in the future.\n",
    "\n",
    "### NISQ algorithm\n",
    "\n",
    "Current (2021) quantum computers are still small in scale and the impact of noise on computational results cannot be ignored. Such quantum computers are distinguished as Noisy Intermediate-Scale Quantum (NISQ) devices [1].  \n",
    "Quantum algorithms that have been theoretically shown to be faster than classical algorithms, such as those described above, cannot be run on NISQ devices at a practical scale and accuracy.  \n",
    "It is said that it will take several decades to develop a quantum computer that can do this (Large scale, fault-tolerant quantum computer).\n",
    "\n",
    "On the other hand, the largest current NISQ devices, as announced as \"quantum transcendence\", have reached a point where they cannot be efficiently simulated by existing classical computers [2].  \n",
    "Therefore, even if we do not wait for several decades, we expect that there will be calculations that can be performed faster than existing classical computers using NISQ devices.  \n",
    "Such an algorithm is called the NISQ algorithm, and a research and development race is currently underway around the world.\n",
    "\n",
    "### On learning NISQ and non-NISQ algorithms\n",
    "In the following tutorials, a distinction will be made between non-NISQ algorithms (\"Universal Quantum Algorithm\") and NISQ algorithms.  \n",
    "The basic construction of a quantum circuit and the quantum gates used are the same for both.  \n",
    "Compared to non-NISQ algorithms, the NISQ algorithm emphasizes the ability to operate with a small number of qubits and to tolerate errors in computation.\n",
    "\n",
    " The non-NISQ quantum algorithms can be better for learning, in point of seeing how quantum computation can be faster than classical computation.　　\n",
    "On the other hand, if you want to focus on the practical usage of quantum computers in the near future, it will be valuable to look into the NISQ algorithms.\n",
    "\n",
    "### References\n",
    "\n",
    "[1] Bharti, Kishor, et al. \"Noisy intermediate-scale quantum (NISQ) algorithms.\" arXiv preprint arXiv:2101.08448 (2021).  \n",
    "[2] Arute, Frank, et al. \"Quantum supremacy using a programmable superconducting processor.\" Nature 574.7779 (2019): 505-51  \n",
    "[3] Nielsen, Chuang \"Quantum Computation and Quantum Information\" Cambridge university press."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "name": "tutorial001_basic_circuit.ipynb",
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   "version": "0.3.2"
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   "language": "python",
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