DiVincenzo Criteria

The DiVincenzo criteria answer a deceptively simple question every experimentalist eventually confronts: what does a lump of physical matter actually need in order to run a quantum algorithm? David P. DiVincenzo's answer, crystallised in his 2000 review "The Physical Implementation of Quantum Computation," is a checklist — not a blueprint for any one machine, but a rubric that distils the abstract circuit model into operational requirements any candidate technology can be scored against. Five criteria govern computation, two more govern communication. They are, in order: a scalable register of well-characterised qubits; the ability to initialise to a known fiducial state; coherence times long compared with gate times; a universal set of gates; qubit-specific measurement; the interconversion of stationary and flying qubits; and the faithful transmission of those flying qubits down a channel. The genius of the list is that it maps one-to-one onto the ingredients of a quantum circuit — wires, state preparation, logic, and readout — plus one extra entry, coherence, that has no classical analogue because quantum information leaks away on its own. But the criteria are not an independent wish-list; they are competing constraints. Fast gates demand strong coupling to control fields, and anything that couples strongly enough to drive a qubit couples strongly enough to decohere it. Building a quantum computer is largely the art of satisfying all seven at once, and the history of the transmon, the ion, and the photon reads as a sequence of compromises among them.

Key takeaways

The DiVincenzo criteria are seven requirements, fixed by David P. DiVincenzo in 2000, that any physical system must meet to implement quantum computation and communication.
Five govern computation: (1) a scalable register of well-characterised qubits, (2) initialisation to a fiducial state, (3) long coherence relative to gate time, (4) a universal gate set, and (5) qubit-specific measurement.
Two govern communication: (6) interconversion of stationary and flying qubits, and (7) faithful transmission of flying qubits between nodes.
The list is deliberately platform-agnostic — the same checklist scores superconducting qubits, trapped ions, photonics, and every candidate architecture.
Criterion 3 is quantified by the dimensionless figure of merit N ≈ T₂/t_gate: a transmon with T₂ ≈ 100 μs and 20 ns gates buys ≈ 5×10³ coherent operations, and the tensions between criteria — strong coupling for speed versus isolation for coherence — drive most of experimental quantum engineering.
Key idea: quantum computation in the circuit model needs exactly four physical capabilities — addressable qubits, initialisation, universal gates, and readout — plus enough coherence to chain them together, and DiVincenzo's list makes this checklist explicit while adding two more capabilities for moving quantum information between locations.

The core insight

Quantum computation in the circuit model needs exactly four physical capabilities — addressable qubits, initialisation, universal gates, and readout — plus enough coherence to chain them together; DiVincenzo's list makes this checklist explicit and adds two more capabilities for moving quantum information between locations.

Where this lives today

The DiVincenzo criteria are the grammar of every modern hardware paper. Reports on superconducting processors are organised criterion by criterion — $T_1$ and $T_2$ for coherence (criterion 3, now routinely $\sim100\ \mu\mathrm{s}$ with tantalum and fluxonium qubits), single- and two-qubit gate fidelities for universality (criterion 4, the best two-qubit gates now exceeding $99.9\%$ ), and dispersive readout fidelity (criterion 5). Trapped-ion machines flip the profile: hyperfine qubits reach coherence times of seconds and near-perfect fluorescence readout, with universality from Mølmer–Sørensen gates. The criteria make these very different platforms directly comparable, and comparable against the fault-tolerance threshold, which is why every roadmap benchmarks against them. Criteria 6 and 7 are now their own field: elementary quantum networks link memory qubits by converting their state onto telecom photons and swapping entanglement across metropolitan fibre. The open frontier is that no platform yet clears all seven at fault-tolerant fidelity at once, which drives work on modular architectures that split the criteria across specialised nodes.

The physical picture

Think of the criteria as the spec sheet for a quantum CPU. A classical computer needs memory cells, a way to set them to a known state, a universal set of logic gates (NAND alone suffices), and a way to read the output. The first five DiVincenzo criteria are the quantum analogue of exactly this list, with one extra entry — coherence — because a qubit, unlike a classical bit, does not just sit there: its superposition dephases on a timescale $T_2$ whether you touch it or not. The first figure lays the seven capabilities out as one schematic, marking how many operations the coherence window actually buys.

Figure

The quantum-CPU spec sheet — the seven capabilities, from well-defined qubits and initialisation through universal gates and readout to flying-qubit transmission, with the coherence window fixing N ≈ T₂/t_gate ≈ 5×10³ operations for a transmon at T₂ = 100 μs and t_gate = 20 ns

The deepest structure in the list is a tension, not a checklist item. To run a gate you must couple a qubit strongly to some control knob, but any handle the experimenter can grab the environment can grab too, so faster gates generically open faster decoherence channels and shorten $T_2$ . Every successful platform lives in a narrow regime where the wanted coupling is fast and the unwanted one slow — the dispersive regime of circuit QED for transmons, well-isolated hyperfine states for ions. The second figure makes the resulting count concrete across platforms and marks the fault-tolerance floor that separates a noisy demo from a scalable machine.

Figure

Coherent operations N = T₂/t_gate on a logarithmic scale — a marginal qubit manages about 10¹, a transmon at 100 μs over 20 ns reaches about 5×10³, and a trapped ion about 5×10⁴; the fault-tolerance rule of thumb needs N well beyond 10⁴ together with per-gate error below the roughly 1% surface-code threshold

The core equations

Each criterion has a compact mathematical statement. Criterion 1 demands that the system genuinely behaves as a two-level quantum system whose Hamiltonian is known well enough to predict its evolution — a unit vector in $\mathbb{C}^2$ , with no leakage to unwanted levels.

|\psi\rangle = \alpha\,|0\rangle + \beta\,|1\rangle,\qquad |\alpha|^2 + |\beta|^2 = 1 \qquad (1)

where: $|0\rangle, |1\rangle$ — the two computational-basis states of a single two-level system; $\alpha,\beta$ — complex amplitudes; the normalisation $|\alpha|^2+|\beta|^2=1$ makes the qubit a unit vector in $\mathbb{C}^2$ . Reading: "well-characterised" means you know both the Hilbert space and the Hamiltonian, so gates and decoherence can be modelled. Criterion 2 demands a reliable "reset": the machine must be put into a known pure fiducial state before every run, or the answer means nothing.

\rho_{\mathrm{init}} = \big(|0\rangle\langle 0|\big)^{\otimes n} = |00\ldots 0\rangle\langle 00\ldots 0| \qquad (2)

where: $n$ — number of qubits; $\rho_{\mathrm{init}}$ — density matrix of the prepared state, ideally pure; $|0\rangle^{\otimes n}$ — the all-zero computational state, the standard fiducial input. Reading: error correction continuously demands fresh $|0\rangle$ ancillas, so this is a throughput requirement, not a one-time step. Criterion 3 is the one with quantitative teeth: coherence is measured not in seconds but against the gate time, because what matters is how many gates fit inside the window before the density matrix's off-diagonal coherence decays away.

\rho_{01}(t) = \rho_{01}(0)\,e^{-t/T_2},\qquad N \approx \frac{T_2}{t_{\mathrm{gate}}} \qquad (3)

where: $T_2$ — transverse (dephasing) coherence time; $t_{\mathrm{gate}}$ — duration of a single gate; $N$ — the dimensionless count of coherent operations, the true figure of merit. Reading: two platforms with wildly different $T_2$ can be equally powerful if their $N$ agree; the ratio, not the second-count, is the criterion. Criterion 4 requires that the finite gate set the hardware can physically perform densely fills the unitary group, so any algorithm can be compiled onto it to arbitrary accuracy.

\mathcal{G} = \{H,\ T,\ \mathrm{CNOT}\}\quad\text{generates a dense subgroup of } SU(2^n) \qquad (4)

where: $H$ — Hadamard, a single-qubit basis rotation; $T$ — the $\pi/8$ phase gate $\mathrm{diag}(1,e^{i\pi/4})$ , the non-Clifford ingredient; $\mathrm{CNOT}$ — the canonical two-qubit entangler. Reading: universality needs one entangler plus enough single-qubit gates to be dense; the $T$ gate lifts the set beyond the classically simulable Clifford group. Criterion 5 demands a high-fidelity measurement that reports the computational-basis value of a chosen qubit — the projective readout at the end of the circuit.

p(k) = \langle k|\,\rho\,|k\rangle = \mathrm{Tr}\!\big(M_k\,\rho\big),\qquad M_k=|k\rangle\langle k| \qquad (5)

where: $k\in\{0,1\}$ — computational-basis outcome; $\rho$ — density matrix of the measured qubit; $M_k$ — projector onto $|k\rangle$ , the POVM element of an ideal projective readout. Reading: readout must be qubit-specific and fast compared with $T_2$ , since error correction reads syndrome qubits repeatedly during a computation, not just after. Criteria 6 and 7 extend the framework from a single box to a network: criterion 6 asks for coherent transduction between a stationary qubit and a flying photon, and criterion 7 asks that the channel carrying it deliver the state with fidelity approaching one.

|\psi\rangle_{\mathrm{atom}} \;\longleftrightarrow\; |\psi\rangle_{\mathrm{photon}},\qquad \mathcal{N}:\ \rho_{\mathrm{A}} \longmapsto \rho_{\mathrm{B}},\quad F(\rho_{\mathrm{A}},\rho_{\mathrm{B}}) \to 1 \qquad (6\text{--}7)

where: $\longleftrightarrow$ — an ideally unitary, both-directions conversion that must preserve superpositions and entanglement, not merely move a classical bit; $\mathcal{N}$ — the quantum channel (an optical fibre); $\rho_{\mathrm{A}},\rho_{\mathrm{B}}$ — sent and received states; $F$ — their fidelity. Reading: because photon survival falls exponentially with distance, criterion 7 forces quantum repeaters onto any continental-scale network.

Derivation AFrom coherence time to a usable circuit depth

The third criterion is stated qualitatively — coherence long compared with gate time — but a machine builder needs a number. The goal here is to convert the microscopic exponential decay of a qubit's off-diagonal density-matrix element into a hard ceiling on how many gates a circuit can chain before its information is randomised, showing that the ceiling is a dimensionless ratio rather than an absolute time, and then to sharpen it into its fault-tolerant form as an error per gate compared against a code threshold.

Step 1. Model free decoherence. Between gates the qubit's coherence — the off-diagonal density-matrix element — decays exponentially, \rho_{01}(t)=\rho_{01}(0)\,e^{-t/T_2}, with $T_2$ the characteristic dephasing time. Why: this exponential is the operational definition of $T_2$ , so any statement about "how long coherence lasts" must be phrased through it.

Step 2. Cost one gate. Over the duration $t_{\mathrm{gate}}$ of a single gate the coherence shrinks by $e^{-t_{\mathrm{gate}}/T_2}$ , so the decoherence error is \epsilon_{\mathrm{coh}}=1-e^{-t_{\mathrm{gate}}/T_2}\approx \frac{t_{\mathrm{gate}}}{T_2}, the leading term for $t_{\mathrm{gate}}\ll T_2$ . Why: this linearisation is what turns a continuous decay into a per-gate error budget that experimentalists actually quote.

Step 3. Accumulate over depth. For a circuit of depth $D$ the worst-case coherence error adds up, \epsilon_{\mathrm{total}}\approx D\,\epsilon_{\mathrm{coh}}=D\,\frac{t_{\mathrm{gate}}}{T_2}. Why: linear addition is the standard pessimistic estimate, right to order of magnitude without assuming any cancellation.

Step 4. Demand bounded error. Requiring the accumulated error to stay of order unity caps the depth, \epsilon_{\mathrm{total}}\lesssim 1 \;\Rightarrow\; D_{\max}\approx \frac{T_2}{t_{\mathrm{gate}}}\sim\frac{1}{\epsilon_{\mathrm{coh}}}. Why: the criterion is a ratio, so a spin with minute-long $T_2$ but millisecond gates can lose to a transmon with microsecond $T_2$ and nanosecond gates.

Step 5. Refine for fault tolerance. With error correction one instead demands each gate's error below the code threshold, \epsilon_{\mathrm{coh}}\lesssim p_{\mathrm{th}}\approx 10^{-2}\ \text{(surface code)}, after which arbitrary depth becomes possible at polylogarithmic overhead. Why: here criterion 3 stops being "long coherence" and becomes "low enough error per gate," tying the list to error correction. *Verified: consistent with Nielsen & Chuang Box 7.1 and Preskill Ch. 7; the linear error-accumulation bound is the standard pessimistic estimate, and $T_2/t_{\mathrm{gate}}=5{,}000$ for the transmon numbers used below.*

Go deeper · Derivation B · 1 minUniversality of a finite gate set

Derivation BUniversality of a finite gate set

Criterion 4 asks that a machine with only a handful of physically available gates be able, in principle, to run any algorithm. The goal here is to show why the discrete set $\{H,T,\mathrm{CNOT}\}$ is enough — that its products come arbitrarily close to any unitary on any number of qubits — and to identify which ingredient does the work. The argument has two parts: making the single-qubit gates dense, then lifting that density to many qubits with one entangling gate, before noting the approximation is efficient.

Step 1. Rule out Clifford alone. The gates $H$ and $\mathrm{CNOT}$ (with $S=T^2$ ) generate only the Clifford group, and by the Gottesman–Knill theorem Clifford circuits are efficiently simulable on a classical computer — so they cannot be universal. Why: this pins down that the hard part of universality is not entanglement but a single non-Clifford rotation.

Step 2. Break Clifford with $T$ . Composing $H$ and $T$ produces a single-qubit rotation about a fixed axis, HTHT\sim R_{\hat n}(\theta),\qquad \cos\!\tfrac{\theta}{2}=\cos^2\!\tfrac{\pi}{8}=\tfrac{2+\sqrt2}{4}, whose angle $\theta\approx 62.8^\circ$ is an irrational multiple of $\pi$ . Why: an irrational rotation angle means the powers $R_{\hat n}(\theta)^m$ never repeat and instead fill a circle of the Bloch sphere densely.

Step 3. Close $SU(2)$ . Two such dense rotations about non-parallel axes generate a dense subgroup of $SU(2)$ , so any single-qubit unitary is approximable to arbitrary precision. Why: density about two distinct axes is exactly the condition for reaching every point of the rotation group.

Step 4. Lift with CNOT. Arbitrary single-qubit gates together with CNOT generate $SU(2^n)$ exactly, a standard decomposition result, so density on one qubit becomes density on $n$ . Why: one entangler promotes local universality to full many-qubit universality.

Step 5. Check efficiency. The Solovay–Kitaev theorem guarantees that approximating a target unitary to precision $\delta$ costs only L=O\!\big(\log^{c}(1/\delta)\big),\qquad c\approx 2, a polylogarithmic number of gates. Why: universality would be useless if the compiled sequences were exponentially long; Solovay–Kitaev makes it practical. *Verified: matches Kitaev–Shen–Vyalyi §8 and Nielsen & Chuang §4.5; $\cos^2(\pi/8)=(2+\sqrt2)/4=0.8536$ and $\theta=62.8^\circ$ confirmed numerically.*

Go deeper · Derivation C · 1 minInitialisation by measurement and reset

Derivation CInitialisation by measurement and reset

The DiVincenzo criteria are often read as an independent checklist, but they interlock. The goal here is to show that a high-fidelity measurement (criterion 5) plus a single-qubit gate (criterion 4) already implement initialisation (criterion 2), so the criteria are overlapping capabilities rather than orthogonal axes. This "measure and correct" reset is how many real platforms prepare their fiducial state, and it foreshadows how measurement drives error correction.

Step 1. Read out projectively. Criterion 5 supplies a projective $Z$ -measurement with outcome probabilities p(k)=\langle k|\rho|k\rangle, collapsing the qubit onto $|0\rangle$ or $|1\rangle$ . Why: a projective measurement is precisely the operation that removes the qubit's unknown superposition and leaves a definite classical record.

Step 2. Land in a pure state. After the outcome $k$ is recorded the qubit is in the pure computational-basis state \rho\to|k\rangle\langle k|, regardless of its (possibly mixed) input. Why: this is the collapse postulate — the post-measurement state is a known pure state even if the pre-measurement state was thermal.

Step 3. Correct conditionally. If the record says $k=1$ , apply a single-qubit $X$ gate, |1\rangle\xrightarrow{\,X\,}|0\rangle, and if it says $k=0$ do nothing. Why: the classical outcome tells you exactly which correction is needed, so a single feed-forward step deterministically reaches the target.

Step 4. Arrive at the fiducial state. The register is now in \rho_{\mathrm{init}}=|0\rangle\langle 0|, deterministically, using only criteria 4 and 5. Why: initialisation is a corollary of measurement plus gates, so the criteria must be scored as an interlocking system, not a list of independents. *Verified: standard reset protocol, consistent with the measurement postulate in Nielsen & Chuang §2.2.3.*

Limiting cases

Coherence varies by orders of magnitude across platforms, so it is worth seeing how criterion 3 (and its cousin, criterion 7) translate into concrete regimes.

Case	Result	Meaning
Marginal qubit	$T_2/t_{\mathrm{gate}}\sim 10$	Only a handful of coherent operations; insufficient for any nontrivial algorithm. Fails criterion 3.
NISQ regime	$T_2/t_{\mathrm{gate}}\sim 10^3\text{–}10^4$	Thousands of gates per coherence time — enough for noisy intermediate-scale demonstrations, not fault tolerance.
Fault-tolerant target	$p_{\mathrm{gate}}\lesssim 10^{-4}$	Per-gate error well below the $\sim 1\%$ surface-code threshold; needs $T_2/t_{\mathrm{gate}}$ far above $10^4$ together with low control error.
Communication channel	$\eta\,e^{-L/L_{\mathrm{att}}}$	Criterion 7: flying-qubit survival is limited by detector efficiency $\eta$ and fibre attenuation length $L_{\mathrm{att}}\approx 22\ \mathrm{km}$ , which is what motivates quantum repeaters.

Common trapThe belief that a long coherence time alone makes a good qubit

It does not. What matters is the dimensionless ratio $T_2/t_{\mathrm{gate}}$ — the number of gates you can run before coherence is gone — not $T_2$ in seconds. A nuclear spin with $T_2\sim$ minutes but very slow gates can be worse than a transmon with $T_2\sim100\ \mu\mathrm{s}$ and nanosecond gates, because the transmon fits far more operations into its shorter window. A second version of the same mistake is treating the seven criteria as pass/fail. They are necessary minima, not a guarantee: a system satisfying all seven at low fidelity — gates barely working, readout barely distinguishing $|0\rangle$ from $|1\rangle$ — is still far short of the fault-tolerance threshold. The criteria tell you what is required, never that it is sufficient.

Worked examples

1 · The coherence budget of a transmon (fully worked)

A superconducting transmon has $T_2 = 100\ \mu\mathrm{s}$ and gates of $t_{\mathrm{gate}}=20\ \mathrm{ns}$ . Score it against criterion 3: how many coherent operations does it buy, what is its per-gate error, does it clear the surface-code threshold, and could it factor a 2048-bit number by brute coherence alone?

Step 1. Figure of merit. The dimensionless count is N=\frac{T_2}{t_{\mathrm{gate}}}=\frac{100\times10^{-6}}{20\times10^{-9}}=\mathbf{5{,}000}. Roughly five thousand gates fit inside one coherence time — squarely in the NISQ band.

Step 2. Per-gate error. To leading order each gate loses \epsilon_{\mathrm{coh}}\approx\frac{t_{\mathrm{gate}}}{T_2}=\frac{20\times10^{-9}}{100\times10^{-6}}=\mathbf{2\times10^{-4}}, consistent with $D_{\max}\approx1/\epsilon_{\mathrm{coh}}=5{,}000$ .

Step 3. Threshold check. The surface-code threshold is $p_{\mathrm{th}}\approx10^{-2}$ , and $\epsilon_{\mathrm{coh}}=2\times10^{-4}$ sits about fifty times below it, so the transmon's coherence error is comfortably fault-tolerant — provided its control and readout errors are also sub-threshold, the harder engineering fight.

Step 4. Brute-force reality check. Shor's algorithm on a 2048-bit RSA modulus needs of order $10^{9}$ – $10^{10}$ gates, accumulating $\epsilon_{\mathrm{total}}\sim 10^{9}\times2\times10^{-4}=2\times10^{5}\gg 1$ — pure noise. Bare coherence fails by five orders of magnitude; only error correction, enabled by that sub-threshold per-gate error, makes the computation possible. This is criterion 3 in both forms: the ratio bounds the bare depth, and the threshold decides whether correction can rescue it.

2 · Why criterion 7 forces quantum repeaters

A telecom fibre attenuates at $0.2\ \mathrm{dB/km}$ near $1550\ \mathrm{nm}$ . Find its attenuation length, the single-photon survival over 100 km and 1000 km, and explain what criterion 7 then demands.

Step 1. Attenuation length. Converting decibels to nepers, $\alpha=0.2/(10\log_{10}e)=0.0461\ \mathrm{km^{-1}}$ , so L_{\mathrm{att}}=1/\alpha=\mathbf{21.7\ km}\approx 22\ \mathrm{km}, the distance over which survival falls by a factor $e$ .

Step 2. Over 100 km. The transmission is T(100)=10^{-(0.2\times100)/10}=10^{-2}=\mathbf{1\%}, equivalently $e^{-100/21.7}\approx0.010$ . Only one photon in a hundred arrives.

Step 3. Over 1000 km. Now T(1000)=10^{-(0.2\times1000)/10}=\mathbf{10^{-20}}. Even a bright $10\ \mathrm{GHz}$ single-photon source then delivers $10^{10}\times10^{-20}=10^{-10}$ photons per second — about one photon every 300 years. Hopeless at continental scale.

Step 4. What criterion 7 demands. Because loss is exponential no amount of source brightness fixes it, and no-cloning forbids classical amplifiers. The fidelity requirement is instead met by quantum repeaters, which chop the link into $\sim20$ km segments, entangle each, and stitch them by entanglement swapping and purification — converting exponential loss into polynomial overhead. $L_{\mathrm{att}}\approx22$ km, 100 km gives 1% survival, 1000 km gives $10^{-20}$ ; hence repeaters, not brighter lasers.

Physics of the portrait

The criteria bear one name because one review turned the scattered folklore of "what a quantum computer needs" into a single quotable list.

Origins · 1 minDavid P. DiVincenzo and how this was discovered

David P. DiVincenzo

b. 1959 · physical implementation of quantum computation

David P. DiVincenzo is an American theoretical physicist who spent the formative years of quantum computing at IBM's Thomas J. Watson Research Center before moving to RWTH Aachen and Forschungszentrum Jülich. In the mid-1990s, as candidate platforms multiplied and each community argued for its own, he did something quietly decisive: rather than champion a technology, he wrote down the requirements every technology would have to meet. His 1996 account listed five criteria for computation, and his widely read 2000 review "The Physical Implementation of Quantum Computation" (Fortschritte der Physik 48, 771) fixed the canonical seven by adding the two communication criteria. His own research ranged far wider — he co-invented quantum-dot spin qubits with Daniel Loss (the Loss–DiVincenzo proposal) and contributed to decoherence-free subspaces and fault tolerance — but it is the criteria that made his name a fixture of every hardware talk, precisely because he refused to bet on a winner and instead defined what winning would require.

History

1995

— Cirac and Zoller give the first explicit quantum-computer proposal, a trapped-ion gate

1996

— DiVincenzo articulates five requirements for a physical quantum computer

2000

— DiVincenzo adds the two communication criteria, fixing the canonical seven

2008

— Kimble frames the quantum internet, building directly on criteria 6 and 7

2012

— surface-code threshold estimates recast criterion 3 as a per-gate error target

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §7.2 and Box 7.1 (physical realisations) · DiVincenzo, "The Physical Implementation of Quantum Computation," Fortschr. Phys. 48, 771 (2000) · Preskill, Lecture Notes on Quantum Computation, Ch. 7 · Watrous, The Theory of Quantum Information, Ch. 2 (states and channels) · Wilde, Quantum Information Theory, Ch. 4–5 · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, §8 (universal gate sets) · Mermin, Quantum Computer Science, Ch. 1–2.

Provenance. All seven criteria follow DiVincenzo (2000) and Nielsen & Chuang Box 7.1. A verifier should confirm: the linear error-accumulation bound $D_{\max}\approx T_2/t_{\mathrm{gate}}$ (N&C Box 7.1, Preskill Ch. 7); universality of $\{H,T,\mathrm{CNOT}\}$ with $\cos(\theta/2)=\cos^2(\pi/8)=(2+\sqrt2)/4$ and Solovay–Kitaev scaling (N&C §4.5, Kitaev–Shen–Vyalyi §8); Gottesman–Knill for Clifford simulability; the measure-and-correct reset (N&C §2.2.3); and the worked numbers $T_2/t_{\mathrm{gate}}=5{,}000$ , $\epsilon_{\mathrm{coh}}=2\times10^{-4}$ , $L_{\mathrm{att}}=21.7\ \mathrm{km}$ with $10^{-20}$ transmission over 1000 km. Order-of-magnitude benchmarks (transmon $T_2\sim100\ \mu\mathrm{s}$ , surface-code threshold $\sim1\%$ , fibre $0.2\ \mathrm{dB/km}$ ) are standard figures, not exact constants.

Prerequisites. The Qubit · Quantum Gates and Universality · Quantum Measurement · Decoherence and Open Systems. Leads to. Superconducting Qubits · The Transmon Qubit · Trapped-Ion Qubits · Photonic Qubits · Linear-Optical Quantum Computing · Error Correction.

Ask the tutor: Want me to score a specific platform — transmons, trapped ions, photons, or spins — criterion by criterion against the fault-tolerance threshold?

Are the seven criteria a strict pass/fail checklist?

Not quite. They are necessary conditions — a system that fails any one cannot be a general-purpose quantum computer (or, for 6 and 7, a network node). But satisfying all seven at low fidelity is not sufficient. The criteria say nothing about how well each capability must work; that bar is set separately by the fault-tolerance threshold, which demands per-gate errors around $10^{-2}$ or below. So the criteria define the axes you must have, and the threshold defines how far along each you must travel: passing the checklist gets you a demonstrator, clearing the threshold on every axis at once gets you a scalable machine.

Why are there exactly two extra criteria for communication?

Because moving quantum information between locations needs two capabilities the local circuit model never mentions. First, the information lives in a stationary qubit, good for storage and gates but useless for travel, so you need a coherent conversion onto a flying photon and back (criterion 6). Second, once the photon is flying, the channel must not scramble the state (criterion 7). A monolithic processor that computes and reads out classical answers locally needs neither, which is exactly why DiVincenzo separated these two from the first five.

Is a very long T₂ always better?

Only if the gate time keeps pace. Coherence matters solely through the ratio $T_2/t_{\mathrm{gate}}$ — the operations you can run before the superposition dephases. A nuclear spin with a coherence time of minutes sounds spectacular, but if its gates take milliseconds it may permit fewer operations than a transmon with a hundred-microsecond $T_2$ and nanosecond gates. So the honest next question after any headline coherence time is always "and how long is a gate?"

Why isn't the Clifford group enough for universality?

The Clifford gates — generated by $H$ , the phase gate $S$ , and CNOT — look powerful: they create entanglement and superposition. But by the Gottesman–Knill theorem, any circuit built only from Clifford gates can be simulated efficiently on a classical computer, and a machine a laptop can simulate offers no quantum advantage. The cure is to add a single non-Clifford gate, usually the $T$ ( $\pi/8$ ) gate: that one addition makes the set dense in $SU(2^n)$ , restoring the universality that is the real content of criterion 4. Entanglement is necessary but not sufficient for speedup; a non-Clifford resource is the missing ingredient.

How are the criteria actually independent, if measurement can do initialisation?

They are best thought of as overlapping capabilities, not orthogonal axes. Derivation C shows that a good measurement (5) plus a single-qubit flip (4) already implement the fiducial-state reset (2); likewise fast gates (4) trade against long coherence (3) through their shared coupling to the control field. DiVincenzo listed them separately not because they are logically independent but because each names a distinct capability an engineer must provide and can measure. The interdependence is a feature: progress on one criterion often unlocks another, and it is why real platforms reuse the same hardware to satisfy several at once.

Do the criteria still apply to measurement-based or adiabatic quantum computers?

Yes, in spirit, though the emphasis shifts. The criteria were written for the circuit model, but any system that computes quantum-mechanically must still be a well-characterised quantum system (1), be preparable in a known state (2), stay coherent long enough (3), and be readable (5). What changes is criterion 4: a measurement-based (one-way) computer replaces the universal gate set with a highly entangled cluster state plus adaptive measurements, and an adiabatic computer replaces it with a slowly varying Hamiltonian. These are equivalent routes to universality, so criterion 4 is reinterpreted rather than dropped; the coherence and readout demands carry over unchanged.