quantity

fidelity

A measure of how close a quantum state or operation is to the ideal target.
90 related articles · 1 formulas

Description

Fidelity is a quantitative measure of how closely a real quantum state or operation matches its ideal, error-free target. In quantum computing, where operations are probabilistic and noise is ever-present, fidelity tells you how 'true' your qubit is after a gate or measurement. A fidelity of 1 means perfect agreement; anything less indicates the presence of errors, decoherence, or imperfect control. It is the central metric for benchmarking quantum hardware, from single-qubit gates to multi-qubit entangling operations.

History

The concept was formalized in the 1990s as quantum error correction and benchmarking matured, with key contributions from Richard Jozsa and others who adapted classical fidelity measures to quantum mechanics.

How it works

For a quantum state, fidelity is defined as the squared overlap between the actual density matrix and the target pure state: F = ⟨ψ_target|ρ_actual|ψ_target⟩. For quantum operations, it is often computed via the average fidelity over all possible input states, or via the entanglement fidelity that accounts for correlations with an ancilla. In practice, researchers estimate fidelity by preparing known states, applying the operation, and performing tomography or randomized benchmarking to extract the error rate.

In practice

Fidelity is used to calibrate and validate quantum processors, guiding error-correction protocols and comparing the performance of different qubit technologies, as seen in recent records with 120 entangled superconducting qubits.

Fun fact

A fidelity of 99.9% sounds impressive, but for a large quantum computation with millions of gates, even that small error accumulates and must be corrected by fault-tolerant schemes.

Concept graph →

Formulas

$$F(\rho, \sigma) = \operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$$
Quantum fidelity measures similarity between two quantum states.
What is in the formula
FvariableQuantum fidelity between states — fidelity
ρvariableDensity matrix of first quantum state — density matrix
σvariableDensity matrix of second quantum state — density matrix
TroperatorSum of diagonal elements of a matrix — trace operator
√operatorPrincipal square root of a positive semidefinite matrix — matrix square root

Related articles · 90

Quantum Physics2026-08-162 minarXiv:2608.15870express
The Quantum Channel That Won't Give Up
In 1996, physicists asked: how much noise can a quantum channel withstand before it stops transmitting information? A phone call where every third word is replaced by random noise is still understandable. Up to what level of interference can the listener still grasp the meaning?
Quantum Physics2026-08-112 minarXiv:2608.10576express
Snowflake Decoder on Ordinary Chips
Quantum computers make mistakes. Each qubit is like a ball balanced on the tip of a pencil: a slight tremor in the hand and it rolls off. To keep the balls from falling, you need someone to notice the wobble and correct it in time.
Quantum Physics2026-08-092 minarXiv:2608.08821express
Quantum Puzzle: How Measurements Assemble Complex States
Imagine a huge puzzle where each piece is a quantum particle. To complete the picture, you don't just fit the pieces together: you need to entangle them in a special way. Usually, this is done by carefully applying operations one after another, like assembling the puzzle strictly by the instructions, row by row.
Quantum Physics2026-08-062 minarXiv:2608.05535express
A Quantum Channel Like a Muddy River
Alice prepares a quantum particle in one of several states and sends it to Bob. The channel the particle travels through isn't perfect: it's noisy, scatters, interacts with the environment. Bob receives the signal and tries to figure out what Alice actually sent. This is the classic "prepare-send-measure" scheme, and there's always a risk of losing something along the way.
Quantum Physics2026-08-042 minarXiv:2608.03944express
Quantum Protection vs. Classical Memory
A quantum computer is like a fragile snowflake resting on a warm palm. The slightest breath of heat or a gentle vibration, and it turns into a droplet. To keep the snowflake alive, you constantly adjust it, restoring its original shape. That's quantum error correction.
Quantum Physics2026-08-031 minarXiv:2608.01788express
Quantum Mpemba effect: how hot cools faster than cold
A cup of hot coffee sometimes freezes faster than cold tap water. This paradox, known as the Mpemba effect, has long been considered a curiosity of classical physics. Now its quantum counterpart has helped test and tune quantum processors — devices that may one day power supercomputers.
Quantum Physics2026-08-032 minarXiv:2608.02773express
A Quantum Code That Isn't Afraid of Errors
A quantum computer is like an orchestra where each musician-qubit easily loses the rhythm. The slightest noise, and the melody of computation turns into cacophony. To keep the orchestra playing in harmony, you need a conductor who instantly spots wrong notes and corrects them without stopping the performance.
Quantum Physics2026-08-022 minarXiv:2608.01111express
A Quantum Cat on Guard for Computing
A quantum computer based on neutral atoms is like a city where couriers, laser tweezers, physically carry packages between addresses. While one courier drags an atom across the entire chip, the others sit idle. This is slow and error-prone. The new architecture replaces couriers with a shared radio wave.
Atomic Physics2026-07-251 minarXiv:2607.23090express
Dancing Atoms Under Optical Tweezers
Two dancers on stage must move in perfect sync, but if one hesitates, the whole routine is ruined. That's roughly how quantum computers with neutral atoms work: two atoms are entangled by putting them into a Rydberg state (as huge as a balloon compared to an ordinary atom), and then you wait for them to 'dance' their part.
Signal Processing2026-07-141 minarXiv:2607.12491express
How to Hear Silence: Seeking Signals by Brightness
Conventional antennas for precise direction finding must work coherently: they capture not just volume, but also the wave's phase at a given moment. The slightest tuning mismatch, and the direction is lost. The new method relies solely on signal strength. It's like finding your way in fog, seeing only the brightness of lights, not their color.
Cryptography2026-07-141 minarXiv:2607.12742express
How the Cloud Secretly Directs a Drone Swarm Without a Leak
A cloud service dishes out commands to a drone swarm without any knowledge of their movements. The signals are encrypted, but each maneuver leaves a barely visible trace in the cipher, like ink bleeding through paper from the pressure of a pen. Over time, these ink stains coalesce into a pattern, and the hidden data is no longer a secret.
Quantum Physics2026-07-122 minarXiv:2607.10707express
Quantum Error Correction Learns on the Fly
A quantum computer is like an orchestra where each musician is a fragile qubit. The slightest noise, and the melody falls out of tune. To keep the harmony, you need a conductor who instantly hears the false notes and brings the instruments back into line.
Quantum Physics2026-07-072 minarXiv:2607.06718express
Dancing on a Carbon Thread: A Diamond Qubit with Millisecond Memory
Building a quantum network node is like teaching a dancer to balance on a carbon thread over an abyss. One leg must spring lightly from the wire, hurling photon-messengers into the external world. The other must grip the rope tightly, guarding precious quantum information.
Quantum Physics2026-07-011 minarXiv:2607.01422express
Quantum Computer Learns to Play Without a False Note
Quantum computing suffers from 'false notes' — errors that arise from the slightest inaccuracy. Now they are eliminated with the precision of tuning a concert grand piano. The idea traces back to the dreams of pioneers — David Deutsch and Peter Shor — of flawless operations.
Quantum Physics2026-06-291 minarXiv:2606.30468express
Entanglement That Can't Be Broken: Photons in Invulnerable Quantum Communication
Quantum entanglement often behaves like a synchronized dance visible from only one angle. Change the way you observe it, and the particles appear disconnected. This limits secure communications. Now physicists have created a pair of photons whose 'dance' is flawless from all perspectives.
Quantum Physics2026-06-211 minarXiv:2606.22376express
Robot Tuner for Quantum Processors
Tuning a quantum processor is monotonous work: you need to calibrate each qubit — a quantum cell — much like a watchmaker adjusting dozens of tiny gears. Humans get tired and make mistakes.
Mesoscale2026-06-192 minarXiv:2606.21580express
Quantum Capacitance Peers Inside a Topological Qubit
Two pendulums hang from the same crossbar. A push, and they start swinging, gradually falling into and out of sync. In a topological qubit, Majorana particles, special excitations that can store quantum information, play the role of these pendulums.
Quantum Physics2026-06-021 minarXiv:2606.02986express
Quantum Physics2026-05-313 minarXiv:2606.01341express
Counting Light to Ten Thousand
You're pouring water into a glass, but you want exactly one million drops — not approximately, but exactly. An ordinary faucet pours a stream where the number of drops always varies. In the quantum world, photons have the same problem: a laser emits light where the number of particles fluctuates like a crowd in a square.
Machine Learning2026-05-201 minarXiv:2605.21164express
Quantum Assistant Detects Rare Frauds
In the ocean of bank transactions, fraud is rarer than neutron stars in space, just 0.1% of all operations. A conventional detector gets used to the clean flow and loses vigilance. Enter Q-SYNTH, a hybrid system with a quantum generator. It creates plausible fake transactions itself, drawing on scraps of real data.
Quantum Physics2026-05-191 minarXiv:2605.19706express
Quantum Droplets: Precision Unattainable
The quantum world is usually drawn with points, but real instruments only see blurry spots. Heisenberg and von Neumann long suspected: precision is unattainable.
Quantum Physics2026-05-142 minarXiv:2605.14656express
Blind Computation: A Quantum Server That Learns Nothing
Quantum computers in the cloud are like a post office: you send a letter, and someone else reads the address and the contents. In ordinary computing, the server sees your data and algorithm. Blind quantum computing is an unopenable envelope: the server does the work but doesn't know what it's doing.
Quantum Physics2026-05-052 minarXiv:2605.03600express
Magic in a Quantum Battery
A quantum battery is not a box of chemicals but a chain of tiny magnets (spins) capable of storing energy. Its main trick: charge many particles at once, using their quantum entanglement. But there's a catch — not all entanglement works.
Quantum Physics2026-05-042 minarXiv:2605.03017express
Quantum Recipe for a Double
A pot of water boils, and molecules break into a chaotic dance. You cannot stop that moment or freeze the dance, but in the quantum world there is a workaround: you can create a 'double' of the system that stores all the information about its thermal state.
Quantum Physics2026-04-292 minarXiv:2604.26791express
Quantum Entanglement on a Chip: 80 km Without Loss
Two mailboxes at opposite ends of a city. Drop a letter in one, and an identical one instantly appears in the other. That's roughly how quantum entanglement works: two particles behave as a single unit, even when separated by a great distance. But until recently, this connection could only be maintained within a single room or across a lab bench.
Chemical Physics2026-04-272 minarXiv:2604.24607express
How a Neural Network Learns from Quantum Chemistry's Mistakes
Quantum calculations are a recipe with perfect ingredients, but nature-the-chef cooks in her own way. Machine-learned models of atoms memorize the recipe and predict material properties in seconds instead of weeks.
Quantum Physics2026-04-272 minarXiv:2604.24854express
Entanglement in the Crosshairs
Quantum entanglement is like an invisible web that connects particles even when they are far apart. In large systems (say, a piece of material) this web can either quickly weave through all the atoms or tangle into a knot and stay put. The first case is called quantum chaos, the second localization.
Quantum Physics2026-04-251 minarXiv:2604.23444express
A Quantum Bridge from Ruby to Glass
A quantum memory node is a tiny diamond with a defect that can store and emit single photons. But its light has a wavelength of 637 nanometers, the same red as a ruby, and is almost completely absorbed after just a few meters of standard optical fiber.
Quantum Physics2026-04-172 minarXiv:2604.16292express
Quantum Physics2026-04-141 minarXiv:2604.12465express
Quantum Conveyor Belt Instead of Shuffling
A quantum computer is like an orchestra where every musician must come in at exactly the right moment. Previously, to play a complex piece, you had to constantly move the instruments around — slow and noisy. The new scheme, called "parity architecture," keeps everyone in place, and the conductor simply changes the notes.
Quantum Physics2026-04-141 minarXiv:2604.13158express
Five Helpers Speed Up Quantum Counting
A neutral-atom quantum computer is like a library where books not only need to be read but also quickly returned to the shelf. Slow readout of results is like a sluggish librarian holding up the entire queue. Scientists have found a way to speed up this process by adding helpers.
Quantum Physics2026-04-092 minarXiv:2604.08672express
A Qubit That Lives Long: 500 Microseconds Instead of 50
In a quantum computer, a qubit is like a tightrope walker: any puff of wind and balance is lost. Errors accumulate, and the computation falls apart. Usually, they are corrected by comparing neighboring qubits, but that's like asking the tightrope walker to glance at his neighbors — extra movements and risk.
Mesoscale2026-04-032 minarXiv:2604.03483express
Quantum microscope twists layers
Think of a stack of paper: when the sheets lie perfectly aligned, light passes through them uniformly. But turn one sheet by a few degrees, and the picture changes. Something similar happens with layers in certain materials: rotate them relative to each other, and their properties change — sometimes dramatically. Spectroscopy isn't the key here, or rather, it's not the method used.
Chemical Physics2026-04-021 minarXiv:2604.02144express
How to Check the Accuracy of Quantum Calculations
When chemists calculate the behavior of molecules on a computer, they use approximate methods because exact calculations are too complex. For light atoms, there are well-established benchmarks, but for heavy atoms, where relativistic effects matter, such benchmarks did not exist. It's like checking a ruler against another ruler rather than against a standard meter.
Quantum Physics2026-04-011 minarXiv:2604.00741express
Randomness from light: a quantum number generator
A laser that usually produces a steady beam starts to tremble near its threshold, like a string about to sound. This trembling is not a defect but quantum noise: atoms inside the laser emit light randomly, without any external cause.
Mesoscale2026-04-011 minarXiv:2604.01063express
18 Qubits on a Single Chip
Eighteen tiny cells line up in two rows on a piece of germanium. Each cell is a quantum dot — a trap for a single electron whose spin acts as a qubit. Previously, such systems could only be assembled in small numbers, but here we have an entire lattice working as a single ensemble.
Quantum Physics2026-03-301 minarXiv:2603.28627express
Quantum Abacus: Cracking Codes with 10,000 Atoms
Instead of millions of rigidly fixed particles — 10,000 atoms, working as beads on a quantum abacus. Each bead can be both zero and one at the same time. But the key is: if an error creeps into the calculation, the atoms can be physically moved: as if you could rearrange the abacus on the fly so that the glitch doesn't ruin the result.
Quantum Physics2026-03-232 minarXiv:2603.22020express
The Quantum World: Reality as a Cloud of Possibilities
Imagine reality as an undeveloped photograph: until the moment of measurement, it exists only as a cloud of probabilities — a superposition of all possible states. Only the act of observation, like developing film, forces a particle to acquire definite properties.
Quantum Physics2026-03-201 minarXiv:2603.19631express
Two Atoms Against Chaos: A Quantum Memory Record
A typical quantum bit is destroyed by the slightest disturbance — heat, magnetic fields. Physicists made a pair of atoms work together so that disorder wouldn't erase the record. Information was encoded not in each atom individually, but in the difference of their states — like the tension of a rope between two buoys.