Updates for #3695 #3709
Updates for #3695 #3709
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Added some details to match the video. More to be pushed and work done before the utility paper replication. I will finish add contents by the end of tomorrow.
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| "where each term $h_i$ acts on one of the $N$ local subsystems of the total Hamiltonian $H$. In the case of indistinguishable elementary particles, it is important to determine whether the system involves fermions or bosons, as this affects the system's statistics and how it must be modeled. In practice, people are often interested in physical systems in which the elements are presumed to be well-separated or labeled, and thus distinguishable, as in spins on a lattice.\n", | ||
| "where each term $H_i$ acts on one of the $N$ local subsystems (like a single particle or a small group of particles) of the total Hamiltonian $H$. In the case of indistinguishable elementary particles, it is important to determine whether the system involves fermions or bosons, where fermions obey the Pauli Exclusion Principle, meaning no two identical fermions can occupy the same quantum states like electrons. Unlike fermions, multiple bosons can exist in the same quantum state, and these difference affects the system's statistics and how it must be modeled.\n", |
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"where each term
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"In practice, people are often interested in physical systems in which the elements are presumed to be well-separated or labeled, and thus distinguishable, as in spins on a lattice.\n",
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| "This system consists of magnetic dipole spins arranged on a lattice, which are treated as distinguishable particles by counting their address. This system is described by the Transverse-Field Ising Model, and its Hamiltonian is constructed from the sum of two parts:\n", |
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address->location
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| "If you have a system with distinguishable particles, like $1/2$ spins fixed on a lattice, which we took a simple look at above, the Hamiltonian is often already written in a language compatible with qubits. The Pauli-Z operator, for instance, naturally describes a spin's up or down, and no special encoding is needed.\n", | ||
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| "When simulating indistinguishable particles of fermions or bosons, it is necessary to apply an encoding transformation. These particles are used to describe within a special mathematical framework called `second quantization`, which tracks the occupation number of each quantum state by introducing `creation` and `annihilation` operators, where the creation operator $\\hat{a}_i^\\dagger$ adds one particle to state $i$ while the annihilation operator $\\hat{a}_i$ removes one particle from state $i$. Based on this second quantization framework, the fermion can be transformed by Bravyi-Kitaev and Jordan-Wigner. Jordan-Wigner transformation defines the fermionic creation operator\n", |
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When simulating indistinguishable fermions or bosons, it is necessary to apply an encoding transformation. These particles are typically described by a special mathematical framework called second quantization, which tracks the occupation number of each quantum state by introducing creation and annihilation operators, where the creation operator
| "|\\phi\\rangle_{HF} = \\prod\\limits_{i=1}^k \\hat{a}_i^\\dagger |0\\rangle ^{\\otimes N} = \\hat{a}_1^\\dagger ... \\hat{a}_k^\\dagger |0_1 ... 0_k ... 0_N\\rangle\n", | ||
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| "Additionally, an easily prepared ansatz with significant overlap to the true ground state can serve as a good initial state for chemistry problems, such as finding the ground state energy.\n", |
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...ground state energy of a molecule.
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| "|\\psi\\rangle = \\sum_{i=0}^{2^n-1}{a_i |c_i\\rangle}=\\hat{O}|0\\rangle^{\\otimes n}\\\\\n", | ||
| "\\text{with} \\qquad \\sum_{i=0}^{2^n-1}{|a_i^2|}=1\n", | ||
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| "However, this process often requires an exponential number of CNOT gates, making it generally resource-intensive. We often focus on preparing initial states for which implementation resource demands are more modest. For example, product states, where qubits are not entangled, can be prepared using only single-qubit operations, significantly reducing complexity." | ||
| "However, this process often requires an exponential number of CNOT gates, making it generally resource-intensive. We often focus on preparing initial states for which the implementation resource demands are more modest. For this reason, we often focus on preparing initial states that are less complex. A common and practical choice is a product state, where qubits are not entangled, can be prepared using only single-qubit operations, significantly reducing the resource demands of the state preparation and the complexity." |
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A common and practical choice is a product state, in which qubits are not entangled, and which can be prepared using only single-qubit operations. This significantly reduces the resource demands of the state preparation and the complexity.
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| "where we have set $\\hbar=1$ by convention.\n", | ||
| "However, building a quantum circuit that directly implements the full operator $e^{-iHt}$ is typically impossible when our Hamiltonian is a sum of different parts, and this is the time we need the Trotterization.\n", |
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sum of different parts. This is where we use trotterization.
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| "Since this series is titled Quantum Computing in Practice, we will also cover the experimental techniques used to improve the quality of the results. One important procedure involves identifying and removing \"bad\" qubits—those with low gate fidelities or short decoherence times—that could significantly impact the experiment’s outcome. Such problematic qubits may arise from poor calibration or interactions with two-level systems (TLS). Removing these qubits alters the hardware’s native topology, effectively changing the lattice on which the system is simulated.\n", | ||
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| "Additionally, we will discuss how to construct the parametrized quantum circuit that implements the system’s time-evolution using trotterization. A key part of this process is identifying entangling layers within the circuit, which play a crucial role in the main error mitigation technique we will introduce later.\n", | ||
| "Additionally, we will discuss how to construct the parametrized quantum circuit that implements the system’s time-evolution using Trotterization. A key part of this process is identifying entangling layers within the circuit, which play a crucial role in the main error mitigation technique we will introduce later.\n", |
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trotterization. It is actually lower-case. It is very confusing to me as well. How can "trotterization" be lower-case, but "Hamiltonian" be capitalized!? It turns out that if it is an object that would historically have been referred to using the creator's exact name, like "Hamilton's matrix", then the resulting noun "Hamiltonian" gets capitalized. If it is a process associated with a person, the new word is just an ordinary word, no capitalization. I only know this because I recently looked it up in frustration. See for example: https://courses.physics.illinois.edu/phys496/sp2017/Lectures/Caps&Acronyms-ClickerQuestions.pdf
I added explanations for the formulas that appear in the video, as I thought they were necessary.
More to be pushed and work done before the utility paper replication. I will finish add contents by the end of tomorrow.