# Alexei Kitaev

Alexei Yurievich Kitaev (Russian: Алексей Юрьевич Китаев; born August 26, 1963) is a Russian-American theoretical physicist.
He is currently a professor of theoretical physics and mathematics at the California Institute of Technology and a permanent member of the Kavli Institute for Theoretical Physics.
Kitaev has received awards for his contributions to the field of quantum mechanics, more specifically quantum computing.

## Life

Kitaev was educated in Russia, graduated from the Moscow Institute of Physics and Technology (1986), and a Ph.D. from the Landau Institute for Theoretical Physics under the supervision of Valery Pokrovsky in 1989.
Kitaev worked as a research associate at the Landau Institute between 1989 and 1998. Between 1999 and 2001, he served as a researcher at Microsoft Research. Since 2002, Kitaev has been a professor at Caltech.
In 2021, Kitaev was elected into the National Academy of Sciences.

## Research
### Quantum algorithms

As one of the early figures in quantum computing, Kitaev introduced several quantum algorithms. These algorithms include the quantum phase estimation algorithm (which serves as a subroutine in several other algorithms such as Shor's algorithm and the quantum counting algorithm), and Kitaev's polynomial-time algorithm for the abelian stabilizer problem and the quantum Fourier transform over arbitrary abelian groups.

### Quantum information formalism

Several of Kitaev's contributions came from introducing formalism to describe notions in quantum information and proving foundational theorems about that formalism. One example is Kitaev's introduction of the complexity classes QMA and QIP, and his proof that the 2-local Hamiltonian problem is QMA-complete.
Another example is Kitaev's formalism for quantum computing using mixed states, which is necessary for analyzing noise, mid-circuit measurements, and other notions in quantum computing. This work was also the origin of the diamond norm, which has become widely used in quantum information.
Kitaev also introduced the formalism for quantum computing using local fermionic modes, which he developed with his graduate student Sergey Bravyi. Kitaev and Bravyi demonstrated that bosonic quantum algorithms and fermionic quantum algorithms are computationally equivalent, in the sense that there are efficient protocols for passing back and forth between the two models.
Kitaev additionally helped to develop the formalism of universal quantum gate sets, and independently proved the Solovay–Kitaev theorem, which shows that any universal quantum gate set can efficiently simulate any other universal quantum gate set.

### Toy models

Kitaev has introduced many toy models in the fields of solid-state physics and quantum mechanics. In his words:
   Throughout my career I have been successful inventing toy models, some simple models that capture important features of a more complex problem.
One of Kitaev's early models was the toric code, which is an exactly solvable lattice model for $\mathbb {Z} _{2}$-gauge theory. This model has since been found to describe several physical situations, such as quantum dimer models and atoms in certain Rydberg blockades. In addition to introducing and solving the toric code, Kitaev made several contributions to analyzing the model such as describing the gapped boundaries of the toric code and describing the phase diagram of the toric code subject to external fields. Kitaev also introduced a generalized version of the toric code, known as Kitaev's quantum double model, which is an exactly solvable lattice model for $G$-gauge theory for any finite group $G$.
Another toy model is Kitaev's honeycomb model, which is an exactly solvable lattice model for topological order with non-abelian anyons. This model is notable not only for being one of the first examples of an explicit Hamiltonian with non-abelian anyons, but also due to the fact that the Hamiltonian is amenable to experimental realization. Many materials have since been found whose effective Hamiltonians are similar to the honeycomb model, and such materials are known as Kitaev spin liquids.
Another example is the Kitaev chain model of a 1D topological superconductor, which demonstrates that unpaired Majorana fermions can be present at the endpoints of a 1D superconducting spin chain. This model has since been used as the basis for many experimental searches for Majorana fermions, and serves as the basis for some approaches to topological quantum computing, such as Microsoft's quantum hardware.
Another toy model is Kitaev's E8 state, which was the first constructed example of an invertible topological state (that is, a topological state with no anyons). It is believed that all invertible topological phases in two dimensions are obtained by stacking copies of the E8 state.
Another example is the Sachdev–Ye–Kitaev (SYK) model, which Kitaev introduced is an exactly solvable model meant to describe some essential features of black-hole physics. Kitaev has made wide-ranging contributions in the development of this model.
Another example is the golden chain, which is a toy model for a spin chain with Fibonacci anyons that Kitaev introduced and solved with his collaborators. This model has served as a paradigmatic prototype for studying anyonic spin chains.

### Topological quantum computing

In 1997, Kitaev proposed the notion of a topological quantum computer. In such a computer, quantum information is stored in delocalized topological quantum numbers, which are inherently resistant to noise and decoherence. The original proposal was based on using sufficiently complex non-abelian anyons in topological order, specifically the anyons in the quantum double model. In later work, jointly with Michael Freedman and Zhenghan Wang, Kitaev demonstrated that topological quantum computation was equivalent to the standard quantum circuit model of quantum computation, which allowed algorithms to be translated back-and-forth between topological quantum field theory and quantum circuits (such as the Aharonov–Jones–Landau algorithm).
Kitaev also made contributions towards topological quantum computation in other models, such as the fractional quantum Hall system and made a proposal for a topological qubit in a superconducting current mirror.

### Classification of topological phases

Kitaev has introduced and proved several classification results for topological order, including Kitaev's periodic table, which gives a classification of topological insulators and superconductors in arbitrary dimensions. Another classification result is the conjectural description of (2+1)-dimensional bosonic topological order in terms of pairs $({\mathcal {C}},c_{-})$ where ${\mathcal {C}}$ is a unitary modular tensor category describing the anyons of the phase and $c_{-}$ is a rational number describing the chiral central charge of the phase. Another classification result is Kitaev's 16-fold way, which gives a periodic description of the anyons in the Kitaev spin liquids based on their Chern numbers modulo 16, which can be rephrased in terms of a classification result for minimal fermionic topological phases.

### Algebraic theory of topological quantum information

One of Kitaev's more prominent contributions was connecting the theory of modular tensor categories, which had been developed in the context of conformal field theory and topological quantum field theory, to describing anyons in topological quantum systems. This has led to wide-ranging applications of category theory to describing topological order. Kitaev also proposed to use of module categories to describe gapped boundaries and domain walls of topological quantum systems.
Kitaev has also connected topological information to algebra by studying the entanglement properties of topological systems. He introduced the notion of topological entanglement entropy jointly with John Preskill, and showed that it can be computed using the algebraic theory of topological order. In a similar spirit, Kitaev and collaborators also studied entanglement in spin chains described by conformal field theory and demonstrated that their entanglement has universal features governed by the central charge of the relevant conformal field theory.

### Fault-tolerant quantum computation

One of Kitaev's most foundational contributions was the toric code, and its close cousin the surface code, which have served as the basis for many theoretical and experimental developments in fault-tolerant quantum computation. Along with his collaborators, Kitaev demonstrated a full protocol for quantum memory in which error detection and correction are both implemented in a fault-tolerant fashion. Since 2021 Kitaev has worked with Google Quantum AI on their experimental effort of realizing fault-tolerant quantum computation, whose efforts have made extensive use of the toric code. One issue with the surface code is that it can only naturally implement Clifford gates fault-tolerantly. To resolve this issue, Kitaev and Sergei Bravyi introduced magic state distillation as a method for obtaining a universal gate set.
Another insight of Kitaev's was the Gottesman-Kitaev-Preskill code, discovered jointly with Daniel Gottesman and John Preskill, which serves as a basis for a model of fault-tolerant continuous-variable quantum computing.
Kitaev independently introduced an algorithm for fault-tolerant universal quantum computation, slightly after the original algorithm of Peter Shor.

## Honors and awards

| Year | Award | Institution | Reason |
| 2008 | MacArthur Fellows Program | MacArthur Foundation | Contributions to the field of quantum computing and quantum physics |
| 2012 | Breakthrough Prize in Fundamental Physics | Breakthrough Prizes Board | For the theoretical development of implemeting quantum memories and fault-tolerant quantum computation |
| 2015 | Dirac Medal (ICTP) | International Centre for Theoretical Physics | For the interdisciplinary contributions in condensed matter systems and applications of these ideas to quantum computing. |
| 2017 | Oliver E. Buckley Prize (with Xiao-Gang Wen) | American Physical Society | For theories of topological order and its consequences in a broad range of physical systems |
| 2024 | Henri Poincaré Prize | International Association of Mathematical Physics | Contributions to the development of quantum computing, the study of quantum many-body systems and quantum information |
| 2024 | Basic Science Lifetime Award | International Congress of Basic Science | Contributions to the development of quantum computing |

## Political positions

In February–March 2022, he signed an open letter by Breakthrough Prize laureates condemning the 2022 Russian invasion of Ukraine.