Linear optical quantum computing with photonic qubits

The Gasparoni et al. (2004) optical CNOT is based on the four-photon logic gate by Pittman et al. (2001).

“The last experiment we consider in some detail is the realization of an optical CNOT by Gasparoni, Pan, Walther, Rudolph, and Zeilinger (2004).The experiment is based on the four-photon logic gate by Pittman et al. (2001) depicted in Fig. 11.”

• cites Gasparoni, Pan, Walther, Rudolph, and Zeilinger (2004) sourcePartly supported

  The paper says its four-photon CNOT scheme was first proposed in reference [15] (p. 1), which the reference list identifies as a Pittman, Jacobs and Franson paper from 2002 (p. 4). This supports the connection to a Pittman-associated scheme, but not the claim that it was based on a Pittman et al. (2001) gate. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites Pittman et al. (2001) sourceSupported

Numerical evidence strongly suggests an upper bound of one third for infinite feed-forward without entangled ancillae.

“Indeed, numerical evidence strongly suggests an upper bound of one third for infinite feed-forward without entangled ancillæ (Scheel et al. 2005).”

• cites Scheel et al. 2005 sourceCould not verify

  The Conclusions and Discussion state that the paper is limited to single-shot gates without conditional feed-forward. The Introduction reports a 1/4 bound without feed-forward, but neither passage addresses numerical evidence for a one-third bound with infinite feed-forward. Searches for that bound and for repeated feed-forward attempts found no supporting passage. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

The NS_N gate defined by Eq. 28 can be implemented with probability 1/N^2.

“can be implemented with probability 1/N 2 [see also Scheel and Audenaert (2005)].”

• cites Scheel and Lütkenhaus (2004) sourcePartly supported

  The Introduction and Special cases section report that the ordinary nonlinear sign-shift gate can be implemented with success probability 1/4, matching 1/N² for the two-photon case. The Conclusion and discussion section says that scaling for generalized nonlinear sign-shift gates was left for future investigation; this paper does not establish the claimed 1/N² probability for general N. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites Eisert (2005) sourcePartly supported

  The paper’s non-linear sign-shift discussion (section labeled Abstract; page unavailable) establishes an achievable optimal probability of 1/4 for N=2. For the three-photon gate, it gives only an upper bound of 1/9. These cases do not establish that the NS_N gate can be implemented with probability 1/N² for arbitrary N. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites Scheel and Audenaert (2005) sourceSupported

The O'Brien et al. (2003) CNOT gate is an implementation of the gate proposed by Ralph et al. (2002a).

“The second experiment we consider is the CNOT gate by O'Brien, Pryde, White, Ralph, and Branning, depicted in Fig. 14 (O 'Brien et al. 2003), which is an implementation of the gate proposed by Ralph et al. (2002a).”

• cites O'Brien et al. 2003 sourceNot checked — no full text of the source
• cites Ralph et al. (2002a) sourcePartly supported

  Section IV (p. 4) proposes a conditional linear-optics CNOT gate, supporting the attribution of a gate proposal to Ralph et al. Our copy does not identify O’Brien et al.’s 2003 experiment or establish that it implemented this particular gate. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

In single-rail qubit encoding, a 50:50 beam splitter can deterministically implement the transformation |1,0> -> |1,0> + |0,1>.

“|1, 0 → |1, 0 + |0, 1 . (15”

• Also stated in: § C. Qubits in linear optics · plain

  “This is a 50:50 beam splitter transformation.”

The maximum number of beam splitters needed to decompose an N port unitary is N(N-1)/2.

“Moreover, this is an efficient process: the maximum number of beam splitters needed is N(N -1)/2.”

• cites Reck et al. (1994) sourceNot checked — no full text of the source

Linear optics alone cannot create maximal polarisation entanglement from single polarised photons in a deterministic manner.

“Therefore, linear optics alone cannot create maximal polarisation entanglement from single polarised photons in a deterministic manner (Kok and Braunstein 2000a).”

• cites Kok and Braunstein 2000aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The lossless beam splitter conserves photon number.

“Since the operator H BS commutes with the total number operator, [H BS , n] = 0, the photon number is conserved in the lossless beam splitter, as one would expect.”

The overall probability of the CZ gate is p_CZ = p_NS^2.

“The overall probability of this CZ gate p CZ = p 2 NS .”

M is unitary if and only if d(ρ) is independent of ρ.

“When we define d(ρ) ≡ Tr QA U(ρ ⊗ σ)U † P k , we find that M is unitary if and only if d(ρ) is independent of ρ.”

The induced operation on the qubits is unitary if and only if T is proportional to the identity.

“The induced operation on the qubits in H Q is then unitary if and only if T is proportional to the identity, or T = Tr A σU † P k U ∝ 1 1 ⇔ d(ρ) = d . (33”

The state α|0 + β|1 can be teleported by applying an n+1-point discrete quantum Fourier transform to the input mode and the first n modes of |t_n, and counting the number of photons m in the output mode; the input state is then teleported to mode n+m of the quantum channel.

“We can then teleport the state α|0 + β|1 by applying an n + 1-point discrete quantum Fourier transform (QFT) to the input mode and the first n modes of |t n , and count the number of photons m in the output mode.The input state will then be teleported to mode n + m of the quantum channel (see Fig. 23).”

For two input qubits in the computational basis, the parity gate induces the transformation |H,H_ab → |H,H_cd, |H,V_ab → |HV,0_cd, |V,H_ab → |0,HV_cd, |V,V_ab → |V,V_cd.

“For two input qubits in the computational basis {|H , |V } this gate induces the following transformation: |H, H ab → |H, H cd , |H, V ab → |HV, 0 cd , |V, H ab → |0, HV cd , |V, V ab → |V, V cd , (29”

To implement a CZ gate with a cross-Kerr medium, the Kerr nonlinearity strength should be tau = pi.

“which means that the strength of the Kerr nonlinearity should be τ = π in order to implement a CZ gate.”

In the example with n = 5 and counting two photons (m = 2), the input mode is teleported to mode 7.

“This means that we teleported the input mode to mode 5 + 2 = 7.”

When gates in a computational circuit succeed only with probability p, the entire calculation using N such gates succeeds with probability p^N.

“When the gates in a computational circuit succeed only with a certain probability p, then the entire calculation that uses N such gates succeeds with probability p N .”

A simple criterion that the N ports and projective measurements must satisfy can be derived.

“We can derive a simple criterion that the N ports and the projective measurements must satisfy (Lapaire et al. 2003).”

• cites Lapaire et al. 2003 sourceNot checked yet

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The conditions in Eq. 34 must be fulfilled for {|χ_k} to be completely distinguishable.

“If the outgoing optical modes are denoted by a j , with corresponding annihilation operators â j , then the set of conditions that have to be fulfilled for {|χ k } to be completely distinguishable are”

Instead of preparing two entangled quantum channels |Φ+, the resource 1⊗U_CZ⊗1|Φ+⊗|Φ+ must be prepared.

“Instead of preparing two entangled quantum channels |Φ + , we now have to prepare the resource 1 1 ⊗ U CZ ⊗ 1 1|Φ + ⊗ |Φ + (see Fig. 22).”

When α = β = γ = δ = 1/√2, the output state of the CZ gate circuit is a maximally entangled state.

“In fact, when we choose α = β = γ = δ = 1/ √ 2, then the output state is a maximally entangled state.”

To keep the amplitudes of the output state equal to those of the input state, the relative amplitudes of the terms in |t_n must be equal.

“In order to keep the amplitudes of the output state equal to those of the input state, the relative amplitudes of the terms in |t n must be equal.”

Two photons in separate spatial modes interacting on a 50:50 beam splitter undergo photon bunching, pairing off together.

“photon bunching: the incoming photons pair off together.This is a strictly quantum-mechanical effect, since classically, the two photons could equally well end up in different output modes.”

No amount of linear optics and bucket detection can lead to perfect, albeit inefficient, single-photon resolution.

“no amount of linear optics and bucket detection can lead to perfect, albeit inefficient, singlephoton resolution (Kok 2003).”

• cites Kok 2003 sourceNot checked yet

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Universal gate operations can be implemented with >99% probability of success with about 10^2 operations.

“It has been estimated that universal gate operations can be implemented with > 99% probability of success with about 10 2 operations (Gilchrist et al. 2005).”

• cites Gilchrist et al. 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Approximate number-resolving detectors can be created using only bucket detectors via detector cascading, with good fidelities when the number of modes is large compared to the average photon number and detector efficiencies are high.

“On the other hand, we can create approximate number-resolving detectors using only bucket detectors via detector cascading.”

• cites Kok and Braunstein 2001 sourceNot checked yet

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• cites Rohde 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A few lines of algebra shows that the remaining q−1 parity qubits are in the state |ψ′L = α|0(n)2...|0(n)q + β|1(n)2...|1(n)q.

“A few lines of algebra shows that the remaining q -1 parity qubits are in the state |ψ ′ L = α|0 (n) 2 . . . |0 (n) q + β|1 (n) 2 . . . |1 (n) q . (89 )”

When the success probability of the measurement of a logical qubit is p, the number of qubits in a tree n is given by n = polylog(1/(1-p)).

“When the success probability of the measurement of a logical qubit is given by p, then the number of qubits in a tree n is given by n = polylog 1 1 -p . (85 )”

Operations Xθ and Z can be performed deterministically on parity-encoded qubits.

“As we have seen in section II.H, we can perform the operations X θ and Z deterministically:”

PDC can yield extremely high fidelities (F > 0.99) because the data is usually obtained via post-selection.

“PDC can yield extremely high fidelities (F > 0.99) because the data is usually obtained via post-selection: we take only those events into account that yield the right number of detector coincidents.”

To store a photon for 100 μs in a fibre has a loss probability of approximately 0.54.

“For example, to store a photon for 100 µs in a fibre has a loss probability of p ≈ 0.54.”

Single-photon sources that create Gaussian wave packets are much more robust to mode mismatching than sources that create Lorentzian wave packets.

“As demonstrated by Rohde, Ralph, and Nielsen (2005) , single-photon sources that create Gaussian wave packets are much more robust to mode mismatching than sources that create Lorentzian wave packets.”

• cites Rohde, Ralph, and Nielsen (2005) sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Phase shifters, beam splitters, and polarisation rotations are sufficient to implement any photonic single-qubit operation.

“These transformations are sufficient to implement any photonic single-qubit operation (Simon and Mukunda 1990).”

• cites Simon and Mukunda 1990 sourceNot checked — no full text of the source

The KLM protocol demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible.

“The protocol by Knill, Laflamme, and Milburn [Nature 409, 46 (2001)] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible.”

• Also stated in: § II. A NEW PARADIGM FOR OPTICAL QUANTUM COMPUTING · boosted

  “In 2000, Knill, Laflamme, and Milburn proved that it is indeed possible to create universal quantum computers with linear optics, single photons, and photon detection (Knill et al. 2001).”

• cites Knill, Laflamme, and Milburn [Nature 409, 46 (2001)] sourceNot checked — no full text of the source
• cites Knill et al. 2001 sourceNot checked — no full text of the source

Polarisation and dual-rail logic are mathematically equivalent.

“This evolution has the same Hamiltonian as the beam splitter, and it formalises the equivalence between the so-called polarisation and dual-rail logic.”

KLM constructed a protocol in which probabilistic two-photon gates are teleported into a quantum circuit with high probability.

“In 2001, Knill, Laflamme, and Milburn (KLM 2001) constructed a protocol in which probabilistic two-photon gates are teleported into a quantum circuit with high probability.”

• cites Knill, Laflamme, and Milburn (KLM 2001) sourceNot checked — no full text of the source

Any unitary transformation of N optical modes can be implemented efficiently with an N port interferometer.

“Reck et al. (1994) demonstrated that the converse is also true, i.e., that any unitary transformation of N optical modes can be implemented efficiently with an N port interferometer .”

• cites Reck et al. (1994) sourceNot checked — no full text of the source

Entanglement generated by changing the definition of subsystems in terms of global field modes is inequivalent to entanglement generated by applying true two-qubit gates to single-photon polarisation or dual-rail qubits.

“Entanglement that is generated by changing the definition of our subsystems in terms of the global field modes is inequivalent to the entanglement that is generated by applying true two-qubit gates to single-photon polarisation or dual-rail qubits.”

It is impossible to implement single-qubit and two-qubit gates deterministically for the same physical representation.

“This implies that we cannot implement single-qubit and two-qubit gates deterministically for the same physical representation.”

In the single-rail representation, single-qubit operations cannot be implemented deterministically with linear optical elements because these transformations do not preserve photon number.

“However, in this representation the single-qubit operations cannot be implemented deterministically with linear optical elements, since these transformations do not preserve the photon number (Paris 2000).”

• cites Paris 2000 sourceNot checked yet

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Subsequent error correction in the quantum circuit is used to bring the error rate down to fault-tolerant levels.

“Subsequent error correction in the quantum circuit is used to bring the error rate down to fault-tolerant levels.”

For linear optical quantum computing, a non-linear component must be added to the scheme.

“For linear optical quantum computing, we typically need the ability to (dis-) entangle field modes.We therefore have to add a non-linear component to our scheme.”

There is no computational advantage in restricting a quantum computer to Clifford group gates and computational-basis projective measurements.

“This means there is no computational advantage in restricting the quantum computer to such circuits.”

At the cost of changing the relative amplitudes and introducing a small error in the teleported output state, the success probability of teleporting a single qubit can be boosted to 1 - 1/n^2 (Franson et al. 2002).

“At the cost of changing the relative amplitudes (and therefore introducing a small error in the teleported output state), the success probability of teleporting a single qubit can then be boosted to 1 -1/n 2 (Franson et al. 2002).”

• cites Franson et al. 2002 sourceNot checked yet

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For an arbitrary input state α|0 + β|1 + γ|2, the KLM NS gate succeeds with probability p_NS = 1/4.

“For an arbitrary input state α|0 + β|1 + γ|2 , this occurs with probability p NS = 1/4.”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

The approach of Cerf, Adami, and Kwiat is not scalable since n-qubit circuits involve 2^n distinct paths.

“As we have seen in section I.D, however, their approach is not scalable since n-qubit circuits involve 2 n distinct paths.”

• cites Cerf, Adami, and Kwiat (1998) sourceNot checked — no full text of the source

The Gasparoni et al. CNOT truth table based on four-fold coincidences shows the operation of the CNOT gate.

“The CNOT truth table for this experiment, based on four-fold coincidences, is shown in Fig. 17.This shows the operation of the CNOT gate.”

• cites Gasparoni et al. (2004) sourceNot checked yet

All two-qubit gates based on projective measurements must be probabilistic.

“All the gates that we have discussed so far are probabilistic, and indeed all two-qubit gates based on projective measurements must be probabilistic.”

Since the CZ gate is part of the Clifford group, it can be commuted through Pauli operators X and Z at the cost of more Pauli operators.

“Since the CZ gate is part of the Clifford group, we can commute it through the Pauli operators X and Z at the cost of more Pauli operators.”

• AssumptionRests on an assumption: The CZ gate is part of the Clifford group.

Any advantage quantum algorithms might have over classical protocols is squandered on retrials or hardware if probabilistic gates are used directly.

“Any advantage that quantum algorithms might have over classical protocols is thus squandered on retrials or on the amount of hardware we need.”

Turchette et al. proposed a method of inducing a phase shift when signal and probe modes of different frequency are both populated by a single polarised photon, using a cavity containing caesium atoms.

“Turchette et al. (1995) proposed a different method of inducing a phase shift when a signal mode s and probe mode p of different frequency are both populated by a single polarised photon.By sending both modes through a cavity containing caesium atoms, they obtain a phase shift that is dependent on the polarisations of the input modes:”

• cites Turchette et al. (1995) sourceNot checked yet

The KLM variant of the Bell measurement always has an arbitrarily small error probability ε.

“The KLM variant of the Bell measurement always has an arbitrarily small error probability ǫ.”

Lloyd showed that almost any two-qubit gate is universal for quantum computing in addition to single-qubit gates.

“As was shown by Lloyd (1995), almost any two-qubit gate is universal for quantum computing (in addition to single-qubit gates)”

• cites Lloyd (1995) sourceNot checked — no full text of the source

A 3-port NS gate with success probability p'_NS = (3 − √2)/7 was proposed by Ralph et al. (2002b), using only two beam splitters.

“First, a 3-port NS gate with only marginally lower success probability p ′ NS = (3 -√ 2)/7 was proposed by Ralph et al. (2002b).This gate uses only two beam splitters (see Fig. 8).”

• cites Ralph et al. (2002b) sourceNot checked yet

Linear optical quantum computing exploits the bosonic commutation relation [a, a†] = 1.

“Indeed, linear optical quantum computing exploits exactly this property, i.e., the bosonic commutation relation [ â, â † ] = 1.”

A CZ gate can be constructed in linear optics using two nonlinear sign (NS) gates.

“A CZ gate can be constructed in linear optics using two nonlinear sign (NS) gates.”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

Combining two parity gates in complementary bases with a maximally entangled ancilla state |Φ+ = (|H,H + |V,V)/√2 yields a CNOT gate with success probability 1/4.

“When two parity gates in complementary bases are combined with a maximally entangled ancilla state |Φ + = (|H, H + |V, V )/ √ 2, a CNOT gate with success probability 1/4 is obtained (Pittmann et al. 2001;Koashi et al. 2001).”

• cites Pittmann et al. 2001Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Koashi et al. 2001 sourceNot checked — no full text of the source

If a suitable storage device is available, |Φ+ can be created with a probabilistic protocol using several trials and the successful event stored.

“If we have a suitable storage device, we do not have to make |Φ + on demand: we can create it with a probabilistic protocol using several trials, and store the output of a successful event in the storage device.”

It is possible to extend operations to general circuits constructed from linear elements, single photon sources, and detectors.

“Instead, it is possible to extend our operations to general circuits that can be constructed from linear elements, single photon sources, and detectors.”

The Gasparoni et al. gate is creating entanglement.

“This clearly shows that the gate is creating entanglement.”

The KLM probabilistic NS gate is a 3-port device including two ancillary modes whose output is measured with perfect photon-number discriminating detectors.

“Their probabilistic NS gate is a 3-port device, including two ancillary modes the output of which is measured with perfect photon-number discriminating detectors (see Fig. 7).”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: The detectors used in the KLM NS gate are perfect photon-number discriminating detectors.

Much larger cross-Kerr nonlinearities of tau approximately 10^-5 can be obtained with electromagnetically-induced transparent materials.

“Much larger cross-Kerr nonlinearities of τ ≈ 10 -5 can be obtained with electromagnetically-induced transparent materials (Schmidt and Imamo ǧlu 1996).”

• cites Schmidt and Imamo ǧlu 1996 sourceNot checked — no full text of the source

Detection of the output ports of the Mach-Zehnder interferometer can reveal whether there was a photon in the signal beam without destroying the signal photon, a quantum non-demolition measurement.

“Moreover, we obtain this information without destroying the signal photon.This is called a quantum non-demolition measurement (Imoto et al. 1985).”

• cites Imoto et al. 1985 sourceNot checked — no full text of the source

Gasparoni et al. showed that an equal superposition of H and V for the control qubit and H for the target qubit generated the maximally entangled state |HH + |VV with a fidelity of 81%.

“In addition, Gasparoni et al. showed that an equal superposition of H and V for the control qubit and H for the target qubit generated the maximally entangled state |HH + |VV with a fidelity of 81%.”

• cites Gasparoni et al. (2004) sourceNot checked yet

The Franson variation yields an output state c_j α|0 + c_{j-1} β|1, where j is known and the c_j are the amplitudes of the modified |t_n.

“the Franson varia- tion will yield an output state c j α|0 + c j-1 β|1 , where j is known and the c j are the amplitudes of the modified |t n .”

• cites Franson et al. 2002 sourceNot checked yet

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In order to do useful quantum computing with probabilistic gates, the probabilistic elements must be taken out of the running calculation.

“In order to do useful quantum computing with probabilistic gates, we have to take the probabilistic elements out of the running calculation.”

The problem of incomplete Bell measurement in linear optics was solved by Knill, Laflamme, and Milburn (2001).

“This is one of the problems of linear optical quantum computing that was solved by Knill, Laflamme, and Milburn (2001).”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

The probability of success of the Gasparoni et al. CNOT gate is 1/4.

“The probability of success of this gate is 1/4.”

• cites Gasparoni et al. (2004) sourceNot checked yet

The general upper bound for such gates is 1/2.

“The general upper bound for such gates was found to be 1/2 (Knill 2003).”

• cites Knill 2003 sourceNot checked yet

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The reconstructed density matrix for the O'Brien et al. CNOT gate clearly indicates that a high-fidelity singlet state has been produced.

“The reconstructed density matrix clearly indicates that a high-fidelity singlet state has been produced.”

• cites O'Brien et al. (2003) sourceNot checked — no full text of the source

Scalable quantum computing can be achieved by making ε smaller than the fault-tolerant threshold.

“We can achieve scalable quantum computing by making ǫ smaller than the faulttolerant threshold.”

• AssumptionRests on an assumption: A fault-tolerant threshold exists such that making the error probability ε smaller than it enables scalable quantum computing.

Physically, the reason deterministic two-qubit gates cannot be constructed in the polarisation and dual-rail representation is that photons do not interact with each other.

“Physically, the reason why we cannot construct deterministic two-qubit gates in the polarisation and dual-rail representation, is that photons do not interact with each other.”

• AssumptionRests on an assumption: Photons do not naturally interact with each other.

Near-deterministic teleportation can be performed with only linear optics and photon counting.

“we have just shown that we can perform near-deterministic teleportation with only linear optics and photon counting.”

Van Loock and Lütkenhaus gave straightforward criteria for the implementation of complete projective measurements with linear optics.

“Van Loock and L ütkenhaus gave straightforward criteria for the implementation of complete projective measurements with linear optics (van Loock and L ütkenhaus 2004).”

• cites van Loock and Lütkenhaus 2004 sourceNot checked yet

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It is possible to perform the NS-gate probabilistically using projective measurements.

“However, it is possible to perform the NS-gate probabilistically using projective measurements.”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

The impossibility proofs for deterministic Bell measurements with linear optics are concerned with exact deterministic Bell measurements.

“The resolution in the paradox lies in the fact that the impossibility proofs are concerned with exact deterministic Bell measurements.”

Linearity of quantum mechanics ensures that if the single-rail state can be teleported, any coherent or incoherent superposition of such a state can also be teleported.

“Linearity of quantum mechanics ensures that if we can teleport this state, we can also teleport any coherent or incoherent superposition of such a state.”

• AssumptionRests on an assumption: Quantum mechanics is linear.

A cross-Kerr nonlinearity of tau approximately 10^-5 is still much too small to implement the gates discussed above.

“However, this value of τ is still much too small to implement the gates we discussed above.”

The KLM teleportation protocol uses a 2n-mode quantum channel |t_n = 1/√(n+1) Σ_{j=0}^n |1_j|0_{n-j}|0_j|1_{n-j}.

“Choose the quantum channel to be the 2n-mode state |t n = 1 √ n + 1 n ∑ j=0 |1 j |0 n-j |0 j |1 n-j , (35”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

The inability to perform a deterministic two-qubit gate such as the CNOT with linear optics alone is intimately related to the impossibility of complete Bell measurements with linear optics.

“The inability to perform a deterministic two-qubit gate such as the CNOT with linear optics alone is intimately related to the impossibility of complete Bell measurements with linear optics (L ütkenhaus et al. 1999;Vaidman and Yoran 1999;Calsamiglia 2002).”

• cites Lütkenhaus et al. 1999 sourceNot checked — no full text of the source
• cites Vaidman and Yoran 1999 sourceNot checked — no full text of the source
• cites Calsamiglia 2002 sourceNot checked yet

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The success probability of an array of n CZ gates of this type can be made p = (1/3)^{n+1}, rather than p = (1/9)^n.

“It was also shown that the success probability of an array of n CZ gates of this type can be made to operate with a probability of p = (1/3) n+1 , rather than p = (1/9) n (Ralph 2004).”

• cites Ralph 2004 sourceNot checked — no full text of the source

When a cross-Kerr medium is placed in one arm of a balanced Mach-Zehnder interferometer, a sufficiently strong phase shift tau can switch the field from one output mode to another.

“When the cross-Kerr medium is placed in one arm of a balanced Mach-Zehnder interferometer, a sufficiently strong phase shift τ can switch the field from one output mode to another (see Fig. 5a).”

The success rate of the KLM teleportation protocol is n/(n+1).

“The success rate of this protocol is n/(n + 1) (where we used that |α| 2 + |β| 2 = 1).”

Without any feed-forward mechanism, the success probability of the NS gate cannot exceed 1/4.

“Without any feed-forward mechanism, the success probability of the NS gate cannot exceed 1/4.”

A destructive CNOT gate was demonstrated by Franson et al. (2003) and O'Brien et al. (2003).

“A destructive CNOT gate was demonstrated by Franson et al. (2003) and O'Brien et al. (2003).”

• cites Franson et al. (2003) sourceNot checked — no full text of the source
• cites O'Brien et al. (2003) sourceNot checked — no full text of the source

The benefit of feed-forward might not outweigh its cost.

“This indicates that the benefit of feed-forward might not outweigh its cost.”

In the Cerf, Adami, and Kwiat protocol, n qubits are represented by a single photon in 2^n different paths.

“Their linear optical protocol can be considered a simulation of a quantum computer: n qubits are represented by a single photon in 2 n different paths.”

• cites Cerf, Adami, and Kwiat (1998, 1999)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The only way photons can directly influence each other is via the bosonic symmetry relation.

“The only way in which photons can directly influence each other is via the bosonic symmetry relation.”

KLM constructed an explicit protocol involving off-line resources, quantum teleportation, and error correction.

“They constructed an explicit protocol, involving off-line resources, quantum teleportation, and error correction.”

• cites Knill et al. 2001 sourceNot checked — no full text of the source

Similar schemes using two ancillary photons have been proposed with success probabilities of 20% and 25%.

“Secondly, similar schemes using two ancillary photons have been proposed (Zou et al. 2002;Scheel et al. 2004).These protocols have success probabilities of 20% and 25%, respectively.”

• cites Zou et al. 2002Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Scheel et al. 2004 sourceNot checked yet

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With a suitable storage device, the teleportation resource can be prepared offline with a probabilistic protocol, leaving no probabilistic elements in the computational circuit.

“Again, with a suitable storage device, this can be done off-line with a probabilistic protocol.There are now no longer any probabilistic elements in the computational circuit.”

The success probability of the teleportation protocol can be made arbitrarily large by increasing the number of modes n.

“We can make the success probability of this protocol as large as we like by increasing the number of modes n.”

A cross-Kerr nonlinearity can be used to create an all-optical CZ gate for photonic qubits.

“It is not hard to see that we can use this mechanism to create an all-optical CZ gate for photonic qubits”

There is no simple two-mode unitary operator that transforms the Franson output state into the original input state without knowledge about α and β.

“There is no simple two-mode unitary operator that transforms this output state into the original input state without knowledge about α and β.”

In 1999, Gottesman and Chuang proposed a teleportation trick that removes the probabilistic gate from the quantum circuit and places it in resources that can be prepared offline.

“In 1999, Gottesman and Chuang proposed a trick that removes the probabilistic gate from the quantum circuit, and places it in the resources that can be prepared offline (Gottesman and Chuang 1999).”

• cites Gottesman and Chuang 1999Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Two-photon polarisation Bell states cannot be created deterministically from single-photon sources, linear optics, and destructive photon detection.

“It is known that these states cannot be created deterministically from single-photon sources, linear optics and destructive photon detection (Kok and Braunstein 2000a).”

• cites Kok and Braunstein 2000aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The CNOT gate on the redundantly encoded qubit can be implemented using q CNOT(p) gates.

“Finally, the CNOT gate on the redundantly encoded qubit can be implemented using q CNOT (p) gates.”

When the probability of a successful teleported gate or addition to a cluster state becomes small, photons in the circuit must be stored for a considerable time while off-line preparation of entangled photons takes place.

“When the probability of a successful (teleported) gate or addition to a cluster state becomes small, the photons that are part of the circuit must be stored for a considerable time while the off-line preparation of entangled photons is taking place.”

Numerical simulations indicate that this method allows for combined detector, source and memory efficiencies of η > 55%.

“Numerical simulations indicate that this method allows for combined detector, source and memory efficiencies of η > 55% (Ralph et al. 2005).”

• cites Ralph et al. 2005 sourceNot checked yet

KLM introduced a logical encoding over two polarisation qubits: |0_L = |HH + |VV and |1_L = |HV + |VH.

“KLM introduced the following logical encoding over two polarisation qubits: |0 L = |HH + |VV |1 L = |HV + |V H (38 )”

• cites KLMNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If the excitation probability amplitudes at each frequency are identical (α_n = β_n), the coincidence probability C is zero.

“If the excitation probability amplitudes at each frequency are identical, α n = β n this quantity is zero.”

Optical switches are still quite lossy.

“Optical switches, however, are still quite lossy (Thew et al. 2002).”

• cites Thew et al. 2002 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

At present, all linear optical quantum computer proposals need some kind of quantum memory.

“At present, all linear optical quantum computer proposals need some kind of quantum memory.”

Using the seven qubit CSS quantum code, the photon loss ǫ in the KLM scheme is allowed to be as large as 1.78% ≤ ǫ ≤ 11.5%, depending on the construction of the entangling gates.

“Using the seven qubit CSS quantum code, the photon loss ǫ in the KLM scheme is allowed to be as large as 1.78% ≤ ǫ ≤ 11.5%, depending on the construction of the entangling gates (Silva et al. 2005).”

• cites Silva et al. 2005 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

For time multiplexing, the fibre length (and hence detection time) must increase exponentially for the technique to work.

“However, the fibre length (and hence the detection time) must increase exponentially for this technique to work.”

In the KLM scheme, a gate teleportation fidelity better than 99% requires detectors with an efficiency η > 0.999987.

“Its effect on the original teleportation component in the KLM protocol was studied by Glancy et al. (2002), who found that in the KLM scheme a gate teleportation fidelity better than 99% requires detectors with an efficiency η > 0.999 987.”

• cites Glancy et al. 2002 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Linear optics can be used to improve the output state of non-ideal single-photon sources.

“When single-photon sources are less than ideal, linear optics might be employed in order to improve the output state.”

• cites Berry et al. 2004Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A proof-of-principle experiment demonstrating optical cluster-state quantum computing with four photons was performed in Vienna (Walther et al. 2005).

“A proof-ofprinciple experiment demonstrating optical cluster-state quantum computing with four photons was performed in Vienna (Walther et al. 2005).”

• cites Walther et al. 2005 sourceNot checked yet

Most of the mode-mismatching effect is due to non-identical photon sources.

“Most of this effect is due to non-identical photon sources, which we discussed in the previous section.”

The two-time correlation function has a zero at T=0, reflecting that the probability to detect a single photon immediately after a single photon detection is zero, as the two pulses are separated in time by |μ-ν|.

“Clearly this has a zero at T = 0 reflecting the fact that the probability to detect a single photon immediately after a single photon detection is zero, as the two pulses are separated in time by |µ -ν|.”

In a slightly altered form, the redundancy encountered in the Browne-Rudolph protocol can be used to make a scalable circuit-based optical quantum computer (Hayes et al. 2004; Gilchrist et al. 2005).

“In fact, in a slightly altered form, the redundancy that we encountered in the Browne-Rudolph protocol can be used to make a scalable circuit-based optical quantum computer (Hayes et al. 2004;Gilchrist et al. 2005).”

• cites Hayes et al. 2004 sourceNot checked yet

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• cites Gilchrist et al. 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the limit N→∞, μ→0 with Ω fixed, we obtain a Poisson distribution for the single photon amplitude.

“In the limit N → ∞, µ → 0 with Ω fixed we obtain a Poisson distribution for the single photon amplitude.”

Given a success probability p, the growth requirement for the redundancy encoding is m > (1 + p)/p.

“From this, we can immediately deduce that, given a success probability p, the growth requirement for the redundancy encoding is m > (1 + p)/p.”

As soon as a photon is lost, the next photon is measured in a diagonal basis, thus disentangling the parity qubit from the other parity qubits.

“Therefore, as soon as a photon is lost the next photon is measured in a diagonal basis, thus disentangling the parity qubit from the other parity qubits.”

Using type-I and type-II fusion gates in creating entanglement, the photon loss can be much higher than in the KLM scheme.

“Using the type-I and type-II fusion gates in creating entanglement, the photon loss can be much higher: In the Browne-Rudolph protocol, a low detection efficiency merely reduces the rate with which the cluster state is created, whereas in the circuitbased model a low detection efficiency requires a higher level of encoding.”

The KLM scheme employs a certain level of error correction to turn high-probability teleported gates into near-deterministic gates, even though all-optical components are ideal.

“We have seen that the KLM scheme employs a certain level of error correction in order to turn high-probability teleported gates into near-deterministic gates, even though all-optical components are ideal.”

The gate Xθ costs 2(q−1) CNOT(p) gates.

“Therefore, the gate X θ “costs” 2(q -1) CNOT (p) gates.”

Using the t1 teleporter on a parity qubit, the probability of success of teleportation is increased from 1/2 to 3/4.

“In this way the probability of success of teleportation is increased from 1/2 to 3/4.”

Quantum interference can be demonstrated using identical pulse shapes triggering different photon sources.

“De Riedmatten et al. demonstrated quantum interference by using identical pulse shapes triggering different photon sources (de Riedmatten et al. 2003).”

• cites de Riedmatten et al. 2003 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Every logical qubit must be constructed with multiple photons, such that photon loss can be recovered from.

“Therefore, every logical qubit must be constructed with multiple photons, such that photon loss can be recovered from.”

Numerical simulations indicate that a detector loss of up to 50% can be corrected.

“Numerical simulations indicate that a detector loss of up to 50% can be corrected (Varnava et al. 2005).”

• cites Varnava et al. 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A detection efficiency of 50% is the absolute minimum for LOQC.

“Hence, a detection efficiency of 50% is the absolute minimum (Barrett 2005).”

• cites Barrett 2005 sourceNot checked yet

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Using multiple copies of a non-ideal single-photon source state, linear optics, and ideal photon detection, the success probability can be increased up to 1/2 but not higher.

“Using multiple copies of ρ, linear optics, and ideal photon detection, one may increase the probability up to p = 1/2, but not higher (Berry et al. 2005).”

• cites Berry et al. 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: In the model of a non-ideal single-photon source, the failure output results in a vacuum state.

Physical beam splitters have a lower bound on absorption amplitude.

“Scheel (2005) showed that there is a lower bound on the absorption amplitude in physical beam splitters.”

• cites Scheel 2005 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

The measurement of the qubit can be retried q times.

“We can therefore retry the measurement of the qubit q times.”

The Spedalieri approach provides a level of error detection because we know we should detect exactly n photons, and detecting m = n photons indicates something went wrong.

“The advantage of this approach is that we know that we should detect exactly n photons.If we detect m = n photons, we know that something went wrong, and this therefore provides us with a level of error detection”

Using magnetic sublevels of the ground state of an atomic ensemble, a weak coherent light pulse has been stored for up to 4 ms with a fidelity of 70%.

“Using the magnetic sublevels of the ground state of an atomic ensemble, Julsgaard et al. (2004) stored a weak coherent light pulse for up to 4 ms with a fidelity of 70%.”

• cites Julsgaard et al. 2004 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

If photon sources are not ideal and there is a substantial number of dark counts in the detectors, quantum information is rapidly lost.

“However, if the photon sources are not ideal and if there is a substantial number of dark counts in the detectors, then we rapidly lose quantum information.”

In the cluster-state approach of Browne and Rudolph, the condition of perfect photon counting can be relaxed given ideal photon sources.

“For example, in the cluster-state approach of Browne and Rudolph, we can relax the condition of perfect photon counting given ideal photon sources.”

Parity encoding of an arbitrary polarisation qubit can be achieved by performing a CNOT gate between the arbitrary qubit and an ancilla qubit prepared in the diagonal state, with the arbitrary qubit as target and the ancilla as control.

“Parity encoding of an arbitrary polarisation qubit can be achieved by performing a CNOT gate between the arbitrary qubit and an ancilla qubit prepared in the diagonal state, where the arbitrary qubit is the target and the ancilla qubit is the control.”

Two-photon quantum interference effects can be demonstrated without relying on identical sources by creating subsequent single-photon states in the same source and using a delay line to interfere the two photons.

“This way, two-photon quantum interference effects are demonstrated without having to rely on iden-tical sources (Santori et al. 2002b).”

• cites Santori et al. 2002bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Varnava, Browne, and Rudolph introduced a code exploiting the property that a cluster state is an eigenstate of every stabiliser generator, and that the eigenvalue of each is known beforehand.

“Varnava, Browne, and Rudolph (2005) introduced a code exploiting the property that a cluster state is an eigenstate of every stabiliser generator, and that the eigenvalue of each is known beforehand (we will assume that all eigenvalues are +1).”

• cites Varnava et al. 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: In the cluster-state photon-loss correction, all stabiliser eigenvalues are +1.

In large optical circuits, photon losses can become substantial.

“In large circuits, these losses can become substantial.”

Repeated nested application of an error-correction protocol (concatenation) can reduce errors to arbitrarily small levels.

“Repeated nested application of the EC protocol (so-called concatenation) can then reduce the errors to arbitrarily small levels.”

When one or more qubits are lost, it is no longer possible to establish the parity.

“However, when one or more qubits are lost it is no longer possible to establish the parity.”

The Xθ gate on the redundantly encoded qubit transforms separable states of parity qubits into highly entangled GHZ states.

“Thirdly, the X θ gate on the redundantly encoded qubit is somewhat problematic, since the gate transforms separable states of parity qubits into highly entangled GHZ states.”

At each higher level of concatenation, corresponding encoded teleportation circuits can be constructed that operate with higher and higher probabilities of success.

“At each higher level of concatenation, corresponding encoded teleportation circuits can be constructed that operate with higher and higher probabilities of success.”

The scaling behaviour of the redundancy encoding growth requirement is exactly the same as the Browne-Rudolph protocol when clusters are grown with the type-II fusion gate.

“This is exactly the same scaling behaviour as the Browne-Rudolph protocol when clusters are grown with the type-II fusion gate.”

When all photons are detected, the qubit is projected onto its logical value.

“When all photons are detected, the qubit is projected onto its logical value.”

Since in LOQC single-qubit measurements amount to photon detection, failing to measure a photon is also a single-qubit failure.

“Since in LOQC the single-qubit measurements amount to photon detection, we have a problem: Failing to measure a photon is also a single-qubit failure.”

A single photon cannot be localised in the same sense that a single massive particle can be localised.

“However it was conclusively demonstrated long ago by Newton and Wigner (1949), and also Wightman (1962), that a single photon cannot be localised in the same sense that a single massive particle can be localised.”

• cites Newton and Wigner (1949) sourceNot checked yet
• cites Wightman (1962) sourceNot checked — no full text of the source

Mode mismatching is probably the most important circuit error.

“Probably the most important circuit error is mode mismatching.”

To achieve good visibility in a Hong-Ou-Mandel test, all single-photon sources must generate identical states.

“Single-photon sources must not only create clean single-photon states, in the sense described above, but all sources must also generate identical states in order to achieve good visibility in a Hong-Ou-Mandel test.”

Different error models typically lead to different levels of robustness.

“Different error models will typically lead to different levels of robustness.”

As the spectrum is bounded from below by n=1, it is not possible to choose amplitudes f_n so that n(t) have arbitrarily narrow support on t∈[0,2π).

“As the spectrum is bounded from below by n = 1, it is not possible to choose the amplitudes f n so that the functions n(t) have arbitrarily narrow support on t ∈ [0, 2π ).”

Only if the two single-photon wave packets are identical do we see a complete cancellation of the coincidence probability.

“In other words, only if the two-single photon wave packets are identical do we see a complete cancellation of the coincidence probability.”

When high-voltage Pockels cells are used, the repetition rate is slow, on the order of 10 kHz.

“In addition, when high-voltage Pockels cells are used, the repetition rate is slow (on the order of 10 kHz).”

When using a bucket detector sensitive over a broad frequency range to herald a single photon, the lack of frequency information in the detector read-out will cause the single-photon state to be mixed.

“When we use a bucket detector that is sensitive over a broad frequency range to herald a single photon in the freely propagating mode, the lack of frequency information in the detector read-out will cause the single-photon state to be mixed.”

Using one parameter to describe both photon absorption and detector efficiency will not have a dramatic effect on the threshold.

“Similarly, using one parameter to describe both photon absorption and detector efficiency will not have a dramatic effect on the threshold (Nielsen 2006).”

• cites Nielsen 2006 sourceNot checked yet

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The noise model adopted by Dawson et al. is a simplification of the physical noise expected in cluster state LOQC.

“Even though this noise model accounts for general noise, and the fault-tolerant threshold puts a bound on its magnitude, it is clearly a simplification of the physical noise that is expected in cluster state LOQC.”

It is possible to construct a quantum computer with linear optics, single-photon sources, and photon detection alone.

“We have seen in this review that it is possible to construct a quantum computer with linear optics, singlephoton sources, and photon detection alone.”

The type-II fusion operator projects the input state onto a maximally entangled Bell state with probability 1/2.

“projects the input state onto a maximally entangled Bell state with probability 1/2.”

The type-I fusion operator determines the parity of the input mode with probability 1/2.

“This operation determines the parity of the input mode with probability 1/2.”

A single-qubit error correction circuit can be constructed with polarising beam splitters, half-wave plates, and Pockels cells.

“In a similar fashion, a single-qubit error correction circuit can be constructed with polarising beam splitters, half-wave plates, and Pockels cells (Kalamidas 2005).”

• cites Kalamidas 2005 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

There is a way to create high-fidelity four-photon GHZ states with lossy bucket detectors and inefficient sources.

“For example, there is a way to create highfidelity four-photon GHZ states with lossy bucket detectors and inefficient sources (Gilchrist 2005).”

• cites Gilchrist 2005Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The study by Dawson et al. shows that scalable quantum computing with the 23-qubit code is possible for a maximum loss probability of γ < 3 × 10−3 and a maximum depolarising probability of ǫ < 10−4.

“The study shows that scalable quantum computing with the 23qubit code is possible for a maximum loss probability of γ < 3 • 10 -3 and a maximum depolarising probability of ǫ < 10 -4 .”

• cites Dawson, Haselgrove, and Nielsen (2006a,b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If perfectly efficient detectors (not necessarily photon-number resolving) are available, deterministic polarisation-flip detection for distributing entanglement can be achieved.

“If we assume the availability of perfectly efficient detectors (not necessarily photon-number resolving), deterministic polarisation-flip detection for distributing entanglement can be achieved (Kalamidas 2004).”

• cites Kalamidas 2004Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: In the deterministic polarisation-flip detection scheme, perfectly efficient detectors (not necessarily photon-number resolving) are available.

In principle, the method works when different parity qubits have different-sized encodings.

“In principle, this method works when different parity qubits have different-sized encodings.”

• AssumptionThe authors state this as an assumption: Every parity qubit is encoded with the same number of photons n.

Since Ralph's circuit is a probabilistic protocol, it cannot naively be inserted in a quantum computing circuit.

“However, since this is a probabilistic protocol, this circuit cannot naively be inserted in a quantum computing circuit.”

An architecture with matter qubits can support scalable quantum computing, even with current realistic components.

“It was shown by Barrett and Kok (2005) that such an architecture can support scalable quantum computing, even with current realistic components.”

• cites Barrett and Kok (2005)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Ralph (2003) proposed a simple demonstration circuit that detects and corrects bit flip errors on a single qubit using the encoded qubit state α|0|0 + β|1|1 and an ancilla qubit |0.

“For example, Ralph (2003) proposed a simple demonstration circuit that detects and corrects bit flip errors on a single qubit using the encoded qubit state α|0 |0 + β|1 |1 and an ancilla qubit |0 .”

• cites Ralph (2003) sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Creating large cluster trees or a sizable redundant encoding will amplify other errors, such as dephasing.

“As we mentioned before, photon loss is not the only error in LOQC, and creating large cluster trees or a sizable redundant encoding in the circuit model will actually amplify other errors, such as dephasing.”

Knill et al. (2000) considered a combination of photon loss and dephasing in the original KLM proposal and found that the accuracy threshold for the optical components in that scheme was higher than 99%.

“Knill et al. (2000) considered a combination of photon loss and dephasing in the original KLM proposal and found that the accuracy threshold for the optical components in that scheme was higher than 99%.”

• cites Knill et al. 2000Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is difficult to produce the superposition states required as resources in squeezed or coherent state schemes.

“The downside is that it is difficult to produce the superposition states that are required as resources in such schemes, although considerable theoretical and experimental progress has been made recently (Lund et al. 2004;Wenger et al. 2004).”

• cites Lund et al. 2004 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

• cites Wenger et al. 2004 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Due to the small efficiency of PDC photon sources, these techniques cannot be considered scalable in a practical sense.

“Due to the small efficiency of PDC photon sources, however, these techniques cannot be considered scalable in a practical sense.”

Knill, Laflamme, and Milburn (2001) overturned the conventional wisdom that a lack of direct photon-photon interactions prohibits scalability.

“Knill, Laflamme, and Milburn (2001) overturned the conventional wisdom that a lack of direct photon-photon interactions prohibits scalability.”

• cites Knill, Laflamme, and Milburn (2001) sourceNot checked — no full text of the source

Bell measurements and fan-out gates become deterministic elements in implementations encoding qubits in squeezed or coherent states of light.

“For example Bell measurements and fan-out gates become deterministic elements (van Enk and Hirota 2002;Jeong et al. 2001).”

• cites van Enk and Hirota 2002 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

• cites Jeong et al. 2001 sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

In the CNOT gate demonstrated by O'Brien et al. (2003), simultaneous detection of a single photon at each of the detectors heralds the successful operation of the gate.

“Simultaneous detection of a single photon at each of the detectors heralds the successful operation of the gate.”

• cites O'Brien et al. (2003) sourceNot checked — no full text of the source

Given a certain noise model and error correcting codes, fault-tolerant thresholds can be derived: the errors must be smaller than the threshold value for concatenated error correction to eliminate them all.

“Given a certain noise model and error correcting codes, we can derive fault-tolerant thresholds: The errors must be smaller than the threshold value for concatenated error correction to eliminate them all.”

These two methods were combined in a fault tolerant, near-deterministic quantum computer architecture.

“Recently, these two methods were combined in a fault tolerant, near-deterministic quantum computer architecture (Lim et al. 2005a).”

• cites Lim et al. 2005aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the near-deterministic teleportation scheme of Knill, Laflamme and Milburn, the input state |phi = alpha|0 + beta|1 is teleported to the mth outgoing mode, where m is the number of detected photons in the measurement of the (n+1)-point quantum Fourier transform.

“The input state |φ = α|0 + β|1 is teleported to the m th outgoing mode, where m is the number of detected photons in the measurement of the (n + 1)-point quantum Fourier transform.”

• cites Knill, Laflamme and MilburnNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Pauli operator X on a lost qubit can be measured by measuring all the adjacent qubits in the Z basis.

“We can measure the Pauli operator X on the shaded (lost) qubit by measuring all the adjacent qubits in the Z basis.”