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Add Bell state example for QClojure
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‎deps.edn‎

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org.lwjgl/lwjgl-stb$natives-windows {:mvn/version "3.3.6"}
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org.lwjgl/lwjgl-stb$natives-macos {:mvn/version "3.3.6"}
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generateme/fastmath {:mvn/version "3.0.0-alpha3"}
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clj-http/clj-http {:mvn/version "3.13.1"}}
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clj-http/clj-http {:mvn/version "3.13.1"}
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org.soulspace/qclojure {:mvn/version "0.21.0"}
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}
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:aliases
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{;; Build the site with `clojure -M:clay -A:markdown`

‎site/db.edn‎

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:affiliation [:cim]}
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{:id :luke-zeitlin
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:name "Luke Zeitlin"
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:affiliation []}
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{:id :ludgersolbach
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:name "Ludger Solbach"
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:url "https://github.com/lsolbach"
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:links [{:icon "github" :href "https://github.com/lsolbach"}]
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:affiliation []}]
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:affiliation

‎src/qclojure/examples/bell_state_circuit.clj‎

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^{:kindly/hide-code true
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:clay {:title "Bell State Circuit"
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:quarto {:author :ludgersolbach
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:draft true
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:type :post
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:date "2025-10-10"}}}
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(ns qclojure.examples.bell-state-circuit
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(:require
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[scicloj.kindly.v4.api :as kindly]
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[scicloj.kindly.v4.kind :as kind]
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[org.soulspace.qclojure.application.visualization :as viz]))
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;; # Bell State Circuit Example
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;; This example demonstrates how to create and visualize a Bell state circuit and how to simulate
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;; a quantum computer executing the circuit, using both an ideal simulator and a hardware simulator,
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;; on classical hardware using QClojure.
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;;
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;; ## What is QClojure?
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;; QClojure is a Clojure library for quantum computing that provides tools to create,
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;; execute, and visualize quantum circuits. It allows users to define quantum circuits using a
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;; high-level, functional programming approach.
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;; QClojure supports various quantum gates, measurements, and state manipulations, making it
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;; suitable for building quantum algorithms and exploring quantum computing concepts.
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;; It comes with a variety of quantum algorithms and provides simulators to run quantum circuits
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;; and algorithms on classical hardware.
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;;
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;; With extensions (e.g. for Amazon Braket), QClojure enables users to run their quantum
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;; circuits on real quantum hardware.
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;;
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;; To use QClojure, you have to include it as a dependency in your Clojure
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;; project. Please use the [latest version](https://clojars.org/org.soulspace/qclojure).
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;;
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;; ## What is a Quantum Computer?
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;; A quantum computer is a type of computing device that leverages the principles of quantum mechanics
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;; to perform computations. Unlike classical computers, which use bits as the basic unit of information
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;; (0s and 1s), quantum computers use quantum bits or qubits. Qubits can exist in multiple states simultaneously
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;; due to a property called superposition. This allows quantum computers to process a vast number of
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;; possibilities at once.
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;; Another key property of quantum computers is entanglement, where the state of one qubit can be
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;; directly related to the state of another, regardless of the distance between them. This phenomenon
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;; enables quantum computers to perform certain calculations much more efficiently than classical computers.
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;;
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;; ## What is a Quantum Circuit?
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;; A quantum circuit is a model for quantum computation in which a computation is represented as a
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;; sequence of quantum gates, which are the quantum analogs of classical logic gates. Quantum circuits
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;; manipulate qubits through these gates to perform operations and transformations on their quantum states.
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;; Quantum circuits are typically visualized using circuit diagrams, where qubits are represented as
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;; horizontal lines and quantum gates as symbols placed along these lines.
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;; Quantum circuits can be used to implement quantum algorithms, which are designed to solve specific
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;; problems more efficiently than classical algorithms. Examples of quantum algorithms include Shor's
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;; algorithm for factoring large numbers and Grover's algorithm for searching unsorted databases.
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;;
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;; ## What is a Quantum Gate?
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;; A quantum gate is a fundamental building block of quantum circuits, analogous to classical logic gates
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;; used in classical computing. Quantum gates manipulate the state of qubits, which are the basic
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;; units of quantum information. Unlike classical bits that can be either 0 or 1,
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;; qubits can exist in a superposition of states, allowing quantum gates to perform complex operations.
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;; Quantum gates are represented as unitary matrices, which ensure that the operations they perform
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;; are reversible.
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;;
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;; Common quantum gates include:
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;; - **Hadamard Gate (H)**: Creates superposition by transforming a qubit from a definite state (|0⟩ or |1⟩)
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;; into an equal superposition of both states.
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;; - **Pauli-X Gate (X)**: Also known as the quantum NOT gate, it flips the state of a qubit (|0⟩ to |1⟩ and vice versa).
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;; - **Pauli-Y Gate (Y)**: Similar to the X gate but also introducess a phase shift.
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;; - **Pauli-Z Gate (Z)**: Introduces a phase flip to the |1⟩ state while leaving the |0⟩ state unchanged.
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;; - **CNOT Gate (Controlled NOT)**: A two-qubit gate that flips the state of the target qubit if the control
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;; qubit is in the state |1⟩. It is essential for creating entanglement between qubits.
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;;
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;; ## What is a Bell State?
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;; A Bell state is a specific quantum state of two qubits that represents the simplest and most
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;; well-known example of quantum entanglement. The Bell states are maximally entangled states
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;; and are used in various quantum information protocols, including quantum teleportation and
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;; superdense coding.
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;; Bell states are fundamental in the study of quantum mechanics and quantum computing,
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;; illustrating the non-classical correlations that can exist between quantum systems.
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;;
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;; There are four different Bell states, but the most commonly referenced one is:
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;; |Φ+⟩ = (|00⟩ + |11⟩) / √2
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;;
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;; This state indicates that if one qubit is measured to be in the state |0⟩, the other qubit will
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;; also be in the state |0⟩, and similarly for the state |1⟩, demonstrating perfect correlation between
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;; the two qubits.
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;; The probability of measuring either |00⟩ or |11⟩ is equal, each with a probability of 0.5,
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;; which means that other combinations like |01⟩ or |10⟩ will never be observed in this state.
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;;
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;; ## Creating the Bell State Circuit
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;; The following code creates a simple quantum circuit that generates a Bell state.
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;; First, we need to require the necessary namespaces from QClojure.
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(require '[org.soulspace.qclojure.domain.state :as state]
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'[org.soulspace.qclojure.domain.circuit :as circuit]
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'[org.soulspace.qclojure.application.visualization :as viz]
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'[org.soulspace.qclojure.adapter.visualization.ascii :as ascii]
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'[org.soulspace.qclojure.adapter.visualization.svg :as svg])
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;; Next, we create a quantum circuit with two qubits and apply the necessary quantum gates
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;; to generate the Bell state. We use the Hadamard gate (H) on the first qubit to create
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;; superposition, followed by a CNOT gate to entangle the two qubits.
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(def bell-state-circuit
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(-> (circuit/create-circuit 2 "Bell State Circuit" "Creates a Bell state.")
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(circuit/h-gate 0)
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(circuit/cnot-gate 0 1)))
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;; We can visualize the circuit as ASCII art for the REPL.
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^kind/code
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(viz/visualize-circuit :ascii bell-state-circuit)
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;; For notebooks and documents, we can also visualize the circuit as SVG.
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^kind/hiccup
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(viz/visualize-circuit :svg bell-state-circuit)
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;; ## Executing the Bell State Circuit
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;; To quickly test the circuit in the REPL, we can use the execute-circuit function
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;; from the circuit namespace.
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(def result (circuit/execute-circuit bell-state-circuit))
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;; The result is a map that contains the final state of the qubits after executing the circuit.
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result
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;; ## Using Simulators to Execute the Circuit
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;; We can also use a quantum backend to execute the circuit with more options.
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;; QClojure provides two different simulator backends: an ideal simulator backend
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;; and a hardware simulator backend.
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;; The ideal simulator simulates the quantum circuit without any noise or errors,
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;; while the hardware simulator simulates the quantum circuit with noise and errors
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;; that are present in real quantum hardware.
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;;
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;; First, we need to require the necessary namespaces for the simulators.
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(require
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'[org.soulspace.qclojure.application.backend :as backend]
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'[org.soulspace.qclojure.adapter.backend.ideal-simulator :as ideal-sim]
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'[org.soulspace.qclojure.adapter.backend.hardware-simulator :as hw-sim])
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;; Let's first use the ideal simulator to execute the Bell state circuit.
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(def ideal-simulator (ideal-sim/create-simulator))
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;; We define some options for the execution, such as the results we want to obtain.
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;; In this case, we want to measure the qubits 100 times (shots).
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(def options {:result-specs {:measurements {:shots 100}}})
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;; Now we can execute the circuit using the ideal simulator and the defined options.
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(def ideal-result
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(backend/execute-circuit ideal-simulator bell-state-circuit options))
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;; The result is a map that contains the measurement results and other information
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;; about the execution.
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ideal-result
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;; We can visualize the frequencies of the measurements obtained from the
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;; ideal simulator as a histogram.
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^kind/hiccup
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(viz/visualize-measurement-histogram :svg (get-in ideal-result [:results :measurement-result :frequencies]))
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;; Now we you the hardware simulator to execute the Bell state circuit.
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;; The hardware simulator simulates the quantum circuit with noise and errors
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;; that are present in real quantum hardware.
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(def hardware-simulator (hw-sim/create-hardware-simulator))
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;; We can also select a specific quantum device to simulate. We choose the
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;; IBM Lagos quantum device for this example. The IBM Lagos is a 7-qubit quantum
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;; computer that is available on the IBM Quantum Experience platform.
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(backend/select-device hardware-simulator :ibm-lagos)
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;; We execute the circuit using the hardware simulator and the defined options.
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(def hardware-result
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(backend/execute-circuit hardware-simulator bell-state-circuit options))
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;; Here is the result of the hardware simulation.
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hardware-result
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;; We can visualize the result of the hardware simulation as a histogram of the
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;; measurement frequencies to compare it with the ideal simulation result.
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^kind/hiccup
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(viz/visualize-measurement-histogram :svg (get-in hardware-result [:results :measurement]))
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;; We results are probabilistic, so we may not get exactly the same results every time we
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;; execute the circuit. However, we should see that the results from the ideal simulator
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;; are closer to the expected Bell state results (|00⟩ and |11⟩ with similar counts) compared to the
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;; hardware simulator, which may show some deviations due to noise and errors.
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;; This demonstrates the impact of quantum noise and errors on the execution of quantum circuits
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;; on real quantum hardware.
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;;
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;; ## Conclusion
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;; In this example, we created a simple quantum circuit that generates a Bell state,
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;; visualized the circuit, and executed it using both an ideal simulator and a hardware
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;; simulator provided by QClojure. We observed the differences in the measurement results
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;; between the two simulators, highlighting the effects of noise and errors in quantum computing.

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