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| 1 | +\documentclass{beamer} |
| 2 | +\usetheme{Madrid} |
| 3 | +\usepackage{amsmath,amssymb,braket} |
| 4 | +\usepackage{tikz} |
| 5 | +\usepackage{quantikz} |
| 6 | +\title{The Jordan–Wigner Transformation} |
| 7 | +\author{Lecture Notes} |
| 8 | +\date{} |
| 9 | + |
| 10 | +\begin{document} |
| 11 | + |
| 12 | +\begin{frame} |
| 13 | +\titlepage |
| 14 | +\end{frame} |
| 15 | + |
| 16 | +\begin{frame}{Outline} |
| 17 | +\tableofcontents |
| 18 | +\end{frame} |
| 19 | + |
| 20 | +\section{Fermionic Operators and Second Quantization} |
| 21 | +\begin{frame}{Fermionic Operators and Second Quantization} |
| 22 | +\begin{itemize} |
| 23 | + \item \textbf{Fermionic modes:} Defined by creation ($a_p^\dagger$) and annihilation ($a_p$) operators for each mode $p=0,\dots,N-1$. These operators obey the {\it canonical anticommutation relations}: |
| 24 | + \[ |
| 25 | + \{a_p, a_q\} = 0, |
| 26 | + \quad |
| 27 | + \{a_p, a_q^\dagger\} = \delta_{pq}, |
| 28 | + \] |
| 29 | + where $\{A,B\}=AB+BA$. (Vacuum: $a_p|\text{vac}\rangle=0$) [oai_citation:0‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=A%20system%20of%20,relations%20have%20the%20following%20consequences). |
| 30 | + \item \textbf{Fock space basis:} The number (occupation) basis is |
| 31 | + $$ |
| 32 | + |n_0,\dots,n_{N-1}\rangle = (a_0^\dagger)^{n_0}(a_1^\dagger)^{n_1}\cdots|\text{vac}\rangle,\quad n_p\in\{0,1\}. |
| 33 | + $$ |
| 34 | + Each $a_p^\dagger$ toggles the occupancy of mode $p$ [oai_citation:1‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=,vectors). |
| 35 | + \item \textbf{Pauli exclusion:} Fermionic operators satisfy $a_p^2=0$; one cannot create two fermions in the same mode [oai_citation:2‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=,in%20the%20same%20mode%20twice). Equivalently, $a_p^\dagger a_p$ has eigenvalues 0 or 1 (occupation number). |
| 36 | +\end{itemize} |
| 37 | +\end{frame} |
| 38 | + |
| 39 | +\section{Anticommutation Relations} |
| 40 | +\begin{frame}{Anticommutation and Consequences} |
| 41 | +\begin{itemize} |
| 42 | + \item \textbf{Canonical Anticommutation:} $\{a_p,a_q\}=0$ and $\{a_p,a_q^\dagger\}=\delta_{pq}$ by definition [oai_citation:3‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=A%20system%20of%20,relations%20have%20the%20following%20consequences). |
| 43 | + \item \textbf{Occupation operators commute:} $[n_p,n_q]=0$ with $n_p=a_p^\dagger a_p$, and each $n_p$ has eigenvalues 0 or 1 [oai_citation:4‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=the%20anticommutation%20relations%20have%20the,following%20consequences). |
| 44 | + \item \textbf{Vacuum state:} There is a unique vacuum $|\text{vac}\rangle$ annihilated by all $a_p$; each $a_p^\dagger$ creates a fermion in mode $p$ (raising its occupation to 1) [oai_citation:5‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=,eigenvector%20of%20all%20the%C2%A0%5C%28a%5E%5Cdagger_p%20a_p). |
| 45 | + \item \textbf{Anticommutation implications:} $a_p^2=0$ (Pauli exclusion) and $(a_p^\dagger)^2=0$. On a basis state, |
| 46 | + $a_p|n_0\ldots 1_p \ldots n_{N-1}\rangle = (-1)^{\sum_{q<p}n_q}|n_0\ldots 0_p\ldots n_{N-1}\rangle$ [oai_citation:6‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=%5C%5B%5Cbegin,1%7D%20%5Crangle%20%26%3D%200%20%5C%2C.%5Cend%7Baligned) (sign from earlier occupations), and annihilation on an empty mode gives 0. |
| 47 | +\end{itemize} |
| 48 | +\end{frame} |
| 49 | + |
| 50 | +\section{Motivation for Mapping Fermions to Qubits} |
| 51 | +\begin{frame}{Why Map Fermions to Qubits?} |
| 52 | +\begin{itemize} |
| 53 | + \item \textbf{Quantum simulation:} We wish to simulate fermionic systems (e.g.\ electrons in molecules or lattice models) on a quantum computer of qubits. To do so, we need an explicit map of fermionic operators to qubit operators that preserves anticommutation [oai_citation:7‡futureofmatter.com](https://futureofmatter.com/assets/fermions_and_jordan_wigner.pdf#:~:text=system%20of%20Fermions,his%20famous%201982%20paper%20on). |
| 54 | + \item \textbf{Spin-fermion equivalence:} The Jordan–Wigner transform shows spin-$\frac12$ chains (qubits) can be mapped to fermions. Conversely, fermionic Hamiltonians can be expressed as qubit (spin) Hamiltonians. For example, 1D spin chains like the Ising/XY models are exactly solvable by mapping to free fermions [oai_citation:8‡en.wikipedia.org](https://en.wikipedia.org/wiki/Jordan%E2%80%93Wigner_transformation#:~:text=The%20Jordan%E2%80%93Wigner%20transformation%20is%20a,operators%20and%20then%20diagonalizing%20in). |
| 55 | + \item \textbf{Electronic structure:} Molecular electronic Hamiltonians (second-quantized fermionic operators) can be encoded into qubit Hamiltonians for quantum chemistry simulations. Each fermionic mode (spin-orbital) corresponds to one qubit under Jordan–Wigner [oai_citation:9‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Jordan,the%20occupation%20of%20one%20qubit). |
| 56 | + \item \textbf{Benchmark models:} Models like the Fermi–Hubbard (strongly correlated electrons on lattice) are prime targets for quantum simulators [oai_citation:10‡nature.com](https://www.nature.com/articles/s41467-022-33335-4?error=cookies_not_supported&code=099d8369-c950-46da-9d82-052da949e39d#:~:text=solution%20of%20physical%20problems%20that,excellent%20benchmark%20for%20quantum%20algorithms). Mapping to qubits allows use of quantum algorithms (VQE, QPE) to study these systems. |
| 57 | +\end{itemize} |
| 58 | +\end{frame} |
| 59 | + |
| 60 | +\section{Jordan–Wigner Transformation: Formalism} |
| 61 | +\begin{frame}{Jordan–Wigner: Operator Mapping} |
| 62 | +The Jordan–Wigner (JW) transform explicitly maps each fermionic mode to a qubit. The annihilation operator $a_p$ is mapped to a product of Pauli matrices on qubits: |
| 63 | +\[ |
| 64 | +a_p \;\mapsto\; \frac{1}{2}\,(X_p + iY_p)\,Z_0 Z_1\cdots Z_{p-1}, |
| 65 | +\] |
| 66 | +and the creation operator $a_p^\dagger$ to its Hermitian conjugate: |
| 67 | +\[ |
| 68 | +a_p^\dagger \;\mapsto\; \frac{1}{2}\,(X_p - iY_p)\,Z_0 Z_1\cdots Z_{p-1}. |
| 69 | +\] |
| 70 | +That is, each fermionic mode $p$ is represented by qubit $p$, with a string of $Z$ operators on all lower-index qubits [oai_citation:11‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=%5C%5B%5Cbegin,aligned). |
| 71 | +\begin{itemize} |
| 72 | + \item This mapping ensures the correct anticommutation: the $Z$-string introduces a sign $(-1)^{\sum_{q<p}n_q}$ when swapping fermions, reproducing Fermi statistics. |
| 73 | + \item In terms of Pauli operators: |
| 74 | + \[ |
| 75 | + a_p = (X_p + iY_p)/2 \prod_{k=0}^{p-1} Z_k, \quad |
| 76 | + a_p^\dagger = (X_p - iY_p)/2 \prod_{k=0}^{p-1} Z_k\,. |
| 77 | + \] [oai_citation:12‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=%5C%5B%5Cbegin,aligned) |
| 78 | +\end{itemize} |
| 79 | +\end{frame} |
| 80 | + |
| 81 | +\begin{frame}{JW: Operator Identities} |
| 82 | +Using the mapping, one can express standard fermionic operators in Pauli form: |
| 83 | +\begin{itemize} |
| 84 | + \item \textbf{Number operator:} |
| 85 | + $$n_p = a_p^\dagger a_p = \frac{I - Z_p}{2},$$ |
| 86 | + since $a_p^\dagger a_p$ has eigenvalue 1 if qubit $p$ is $|1\rangle$ and 0 if $|0\rangle$. |
| 87 | + \item \textbf{Fermionic parity:} |
| 88 | + $$(-1)^{\sum_{q<p}n_q} = \prod_{k=0}^{p-1} Z_k,$$ |
| 89 | + implemented by the string of $Z$'s in the mapping. |
| 90 | + \item \textbf{Spin operators:} The Pauli-$Z$ on qubit $j$ corresponds to $2a_j^\dagger a_j - I$ (local fermion number parity) [oai_citation:13‡futureofmatter.com](https://futureofmatter.com/assets/fermions_and_jordan_wigner.pdf#:~:text=operators%20in%20terms%20of%20the,for%20Xj%20by%20noting%20that). For nearest-neighbor products: |
| 91 | + \[ |
| 92 | + X_j X_{j+1} + Y_j Y_{j+1} |
| 93 | + \;=\; a_j^\dagger a_{j+1} + a_{j+1}^\dagger a_j, |
| 94 | + \] |
| 95 | + showing how hopping terms map to two-qubit terms. |
| 96 | + \item \textbf{Inverse transform:} The Pauli operators can be expressed in terms of $a,a^\dagger$. For example, |
| 97 | + \[ |
| 98 | + Z_j = a_j a_j^\dagger - a_j^\dagger a_j, |
| 99 | + \] |
| 100 | + \[ |
| 101 | + X_j = - (Z_0 Z_1 \cdots Z_{j-1})(a_j + a_j^\dagger), |
| 102 | + \] |
| 103 | + \[ |
| 104 | + Y_j = i(Z_0 Z_1 \cdots Z_{j-1})(a_j^\dagger - a_j), |
| 105 | + \] |
| 106 | + which one can verify yields the usual Pauli matrices [oai_citation:14‡futureofmatter.com](https://futureofmatter.com/assets/fermions_and_jordan_wigner.pdf#:~:text=operators%20in%20terms%20of%20the,for%20Xj%20by%20noting%20that). |
| 107 | +\end{itemize} |
| 108 | +\end{frame} |
| 109 | + |
| 110 | +\section{Example: 2-Mode Fermionic Hamiltonian} |
| 111 | +\begin{frame}{Example: 2-Mode Hamiltonian} |
| 112 | +Consider a two-mode fermionic Hamiltonian (e.g.\ tight-binding or Hubbard dimer): |
| 113 | +\[ |
| 114 | +H = \varepsilon\,(a_0^\dagger a_0 + a_1^\dagger a_1) + t\,(a_0^\dagger a_1 + a_1^\dagger a_0). |
| 115 | +\] |
| 116 | +Under Jordan–Wigner: |
| 117 | +\[ |
| 118 | +a_0^\dagger a_0 = \frac{I - Z_0}{2}, |
| 119 | +\quad |
| 120 | +a_1^\dagger a_1 = \frac{I - Z_1}{2}, |
| 121 | +\] |
| 122 | +\[ |
| 123 | +a_0^\dagger a_1 + a_1^\dagger a_0 |
| 124 | += \frac{1}{2}(X_0X_1 + Y_0Y_1). |
| 125 | +\] |
| 126 | +Thus the qubit Hamiltonian becomes |
| 127 | +\[ |
| 128 | +H_{q} = \frac{\varepsilon}{2}(2I - Z_0 - Z_1) \;+\; \frac{t}{2}(X_0X_1 + Y_0Y_1). |
| 129 | +\] |
| 130 | +All terms are now products of Pauli operators acting on qubits 0 and 1. (This illustrates the mapping of on-site energies to $Z$ and hopping to $XX+YY$ interactions.) |
| 131 | +\end{frame} |
| 132 | + |
| 133 | +\section{Circuit Representations} |
| 134 | +\begin{frame}{Circuit Representation of JW Operators} |
| 135 | +Fermionic operators mapped to Pauli operators can be implemented by quantum circuits: |
| 136 | +\begin{itemize} |
| 137 | + \item \textbf{Single-mode phases:} A term like $Z_p$ (from number operators) is a simple single-qubit phase-flip gate. |
| 138 | + \item \textbf{Hopping terms (XX+YY):} For example, the two-qubit interaction |
| 139 | + $e^{-i\frac{\theta}{2}(X_0X_1 + Y_0Y_1)}$ can be realized by standard gate decompositions. One method: |
| 140 | + apply Hadamard gates to bring $X\!X$ coupling into a $Z\!Z$ form on a basis, perform a controlled-$Z$ rotation, and undo the basis change. |
| 141 | + \item \textbf{RYY / RXX gates:} Modern frameworks include gates like $R_{XX}(\theta)=e^{-i\theta X\otimes X}$ and $R_{YY}(\theta)=e^{-i\theta Y\otimes Y}$ [oai_citation:15‡docs.quantum.ibm.com](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RYYGate#:~:text=A%20parametric%202,rotation%20about%20YY). For instance, IBM Qiskit has an RYY gate implementing $e^{-i\theta\,Y\otimes Y}$ [oai_citation:16‡docs.quantum.ibm.com](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RYYGate#:~:text=A%20parametric%202,rotation%20about%20YY). |
| 142 | + \item \textbf{Circuit example:} To implement $e^{-i(\phi/2)X_0X_1}$, use two Hadamards and a controlled-$Z$ rotation: |
| 143 | + \[ |
| 144 | + \Qcircuit @C=1em @R=0.7em { |
| 145 | + \lstick{\ket{q_0}} & \gate{H} & \ctrl{1} & \gate{H} & \qw \\ |
| 146 | + \lstick{\ket{q_1}} & \qw & \targ & \gate{R_z(2\phi)} & \qw |
| 147 | + } |
| 148 | + \] |
| 149 | + (Similar constructions exist for $X\otimes X + Y\otimes Y$ by combining rotations.) |
| 150 | +\end{itemize} |
| 151 | +\end{frame} |
| 152 | + |
| 153 | +\section{Applications in Quantum Simulation} |
| 154 | +\begin{frame}{Applications: Quantum Simulations} |
| 155 | +\begin{itemize} |
| 156 | + \item \textbf{Quantum Chemistry:} Electronic structure Hamiltonians (in second quantization) map directly to qubit Hamiltonians via JW. Each molecular orbital/spin-orbital is a fermionic mode→ qubit [oai_citation:17‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Jordan,the%20occupation%20of%20one%20qubit). For example, Qiskit Nature uses JW for initial mappings of small molecules. |
| 157 | + \item \textbf{Spin Chains and Lattice Fermions:} The JW transform is classic for solving 1D spin chains (Ising, XY) by mapping to free fermions [oai_citation:18‡en.wikipedia.org](https://en.wikipedia.org/wiki/Jordan%E2%80%93Wigner_transformation#:~:text=The%20Jordan%E2%80%93Wigner%20transformation%20is%20a,operators%20and%20then%20diagonalizing%20in). Conversely, simulating lattice fermion models (e.g.\ Fermi–Hubbard) on a qubit array uses JW to encode creation/annihilation operators. The 1D Fermi–Hubbard model is a common benchmark for quantum computers [oai_citation:19‡nature.com](https://www.nature.com/articles/s41467-022-33335-4?error=cookies_not_supported&code=099d8369-c950-46da-9d82-052da949e39d#:~:text=solution%20of%20physical%20problems%20that,excellent%20benchmark%20for%20quantum%20algorithms). |
| 158 | + \item \textbf{Tapering via Symmetry:} In some cases, e.g.\ systems with fixed particle number or spin parity, the JW-mapped Hamiltonian has symmetries (global $Z$-parities) that allow qubit reduction (tapering) [oai_citation:20‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Parity%20mapping%20is%20the,is%20delocalized%20over%20all%20qubits). |
| 159 | +\end{itemize} |
| 160 | +\end{frame} |
| 161 | + |
| 162 | +\section{Other Fermion-to-Qubit Mappings} |
| 163 | +\begin{frame}{Alternative Mappings (Parity, Bravyi–Kitaev)} |
| 164 | +\begin{itemize} |
| 165 | + \item \textbf{Parity mapping:} A variant of JW where parity information is stored locally on qubits. In parity mapping, occupation information is delocalized, and the qubit index encodes parity of lower modes [oai_citation:21‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Parity%20mapping%20is%20the,is%20delocalized%20over%20all%20qubits). This mapping often enables qubit tapering by conserving total particle number. |
| 166 | + \item \textbf{Bravyi–Kitaev (BK):} A mapping that balances locality between JW and parity. BK uses a binary-tree structure to achieve $O(\log N)$ Pauli-weight for parity and update operations [oai_citation:22‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=already%20supports%20the%20following%20fermionic,mappings). It generally reduces the number of gates for some operators at the cost of more complex index management. |
| 167 | + \item \textbf{Comparison:} |
| 168 | + JW is simplest (long $Z$-strings, $O(N)$ weight), parity and BK trade off Pauli string lengths differently. The best choice depends on hardware connectivity and symmetry exploitation [oai_citation:23‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=already%20supports%20the%20following%20fermionic,mappings) [oai_citation:24‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Parity%20mapping%20is%20the,is%20delocalized%20over%20all%20qubits). |
| 169 | +\end{itemize} |
| 170 | +\end{frame} |
| 171 | + |
| 172 | +\section{References} |
| 173 | +\begin{frame}[allowframebreaks]{References} |
| 174 | +\footnotesize |
| 175 | +\begin{thebibliography}{10} |
| 176 | +\bibitem M.\ Nielsen, “The Fermionic CCRs and the Jordan–Wigner Transform,” {\it arXiv:}quant-ph/0205030 (2005) [oai_citation:25‡futureofmatter.com](https://futureofmatter.com/assets/fermions_and_jordan_wigner.pdf#:~:text=system%20of%20Fermions,his%20famous%201982%20paper%20on) [oai_citation:26‡futureofmatter.com](https://futureofmatter.com/assets/fermions_and_jordan_wigner.pdf#:~:text=operators%20in%20terms%20of%20the,for%20Xj%20by%20noting%20that). |
| 177 | +\bibitem I.\ S.\ Wang {\it et al.}, {\it OpenFermion tutorials} (Google Quantum AI), *Jordan–Wigner and Bravyi–Kitaev Transforms* [oai_citation:27‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=A%20system%20of%20,relations%20have%20the%20following%20consequences) [oai_citation:28‡quantumai.google](https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms#:~:text=Under%20the%20Jordan,to%20qubit%20operators%20as%20follows). |
| 178 | +\bibitem A.\ Moll {\it et al.}, “Fermionic mappings in electronic structure,” Qiskit Nature Tutorial (2022) [oai_citation:29‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Jordan,the%20occupation%20of%20one%20qubit) [oai_citation:30‡qiskit-community.github.io](https://qiskit-community.github.io/qiskit-nature/tutorials/06_qubit_mappers.html#:~:text=The%20Parity%20mapping%20is%20the,is%20delocalized%20over%20all%20qubits). |
| 179 | +\bibitem Wikipedia, *Jordan–Wigner transformation*, (provides background and context) [oai_citation:31‡en.wikipedia.org](https://en.wikipedia.org/wiki/Jordan%E2%80%93Wigner_transformation#:~:text=The%20Jordan%E2%80%93Wigner%20transformation%20is%20a,operators%20and%20then%20diagonalizing%20in). |
| 180 | +\bibitem Qiskit Documentation, *RYYGate*, describes $e^{-i\theta Y\otimes Y}$ implementations [oai_citation:32‡docs.quantum.ibm.com](https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RYYGate#:~:text=A%20parametric%202,rotation%20about%20YY). |
| 181 | +\bibitem S.\ Stanisic {\it et al.}, “Observing Fermi–Hubbard ground state properties,” *Nat.\ Commun.* 13, 5743 (2022) [oai_citation:33‡nature.com](https://www.nature.com/articles/s41467-022-33335-4?error=cookies_not_supported&code=099d8369-c950-46da-9d82-052da949e39d#:~:text=solution%20of%20physical%20problems%20that,excellent%20benchmark%20for%20quantum%20algorithms). |
| 182 | +\end{thebibliography} |
| 183 | +\end{frame} |
| 184 | + |
| 185 | +\end{document} |
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