Quantum logic gate
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In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits . They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.
Unlike many classical logic gates, quantum logic gates are reversible . However, it is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancillary bits . The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.
Contents
- 1 Representation
- 2 Notable examples
- 2.1 Hadamard (H) gate
- 2.2 Pauli-X gate
- 2.3 Pauli-Y gate
- 2.4 Pauli-Z (
- 2.5 Squares of a Pauli Matrix Are the Identity Matrix
- 2.6 Square root of NOT gate (√NOT)
- 2.7 Phase shift (
- 2.8 Swap (SWAP) gate
- 2.9 Square root of Swap gate (√SWAP)
- 2.10 Controlled (cX cY cZ) gates
- 2.11 Toffoli (CCNOT) gate
- 2.12 Fredkin (CSWAP) gate
- 2.13 Ising (XX) gate
- 2.14 Deutsch (
- 3 Universal quantum gates
- 4 Measurement
- 5 Circuit composition and entangled states
- 6 History
- 7 See also
- 8 Notes
- 9 References
Representation[ edit ]
Quantum logic gates are represented by unitary matrices . The number of qubits in the input and output of the gate must be equal; a gate which acts on
$$\displaystyle n
qubits is represented by a
$$\displaystyle 2^n\times 2^n
unitary matrix. The quantum states that the gates act upon are vectors in
$$\displaystyle 2^n
complex dimensions. The base vectors are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on spaces of one or two qubits, just like the common classical logic gates operate on one or two bits.
The vector representation of a single qubit is:
The vector representation of two qubits is:
The action of the gate on a specific quantum state is found by multiplying the vector
$$\displaystyle
which represents the state by the matrix
$$\displaystyle U
representing the gate.
Notable examples[ edit ]
Hadamard (H) gate[ edit ]
The Hadamard gate acts on a single qubit. It maps the basis state
$$0\rangle
to
$$\displaystyle \frac 1\rangle \sqrt 2
and
$$\displaystyle
to
$$\displaystyle \frac 1\rangle \sqrt 2
, which means that a measurement will have equal probabilities to become 1 or 0 (i.e. creates a superposition ). It represents a rotation of
$$\displaystyle \pi
about the axis
$$\displaystyle (\hat x+\hat z)/\sqrt 2
. Equivalently, it is the combination of two rotations,
$$\displaystyle \pi
about the Z-axis followed by
$$\displaystyle \pi /2
about the Y-axis. It is represented by the Hadamard matrix :
Circuit representation of Hadamard gate
The Hadamard gate is the one-qubit version of the quantum fourier transform .
Since
$$\displaystyle HH^*=I
where I is the identity matrix, H is indeed a unitary matrix .
Pauli-X gate[ edit ]
Quantum circuit diagram of a NOT-gate
The Pauli-X gate acts on a single qubit. It is the quantum equivalent of the NOT gate for classical computers (with respect to the standard basis
$$\displaystyle
,
$$\displaystyle
, which privileges^{[ clarification needed ]} the Z-direction) . It equates to a rotation of the Bloch sphere around the X-axis by
$$\displaystyle \pi
radians. It maps
$$0\rangle
to
$$\displaystyle
and
$$\displaystyle
to
$$\displaystyle
. Due to this nature, it is sometimes called bit-flip . It is represented by the Pauli matrix :
Pauli-Y gate[ edit ]
The Pauli-Y gate acts on a single qubit. It equates to a rotation around the Y-axis of the Bloch sphere by
$$\displaystyle \pi
radians. It maps
$$\displaystyle
to
$$1\rangle
and
$$1\rangle
to
$$0\rangle
. It is represented by the Pauli Y matrix:
Pauli-Z (
The Pauli-Z gate acts on a single qubit. It equates to a rotation around the Z-axis of the Bloch sphere by
$$\displaystyle \pi
radians. Thus, it is a special case of a phase shift gate (which are described in a next subsection) with
$$\displaystyle \phi =\pi
. It leaves the basis state
$$0\rangle
unchanged and maps
$$1\rangle
to
$$\displaystyle –
. Due to this nature, it is sometimes called phase-flip. It is represented by the Pauli Z matrix:
Squares of a Pauli Matrix Are the Identity Matrix[ edit ]
Note the square roots of the identity matrix
Square root of NOT gate (√NOT)[ edit ]
Quantum circuit diagram of a square-root-of-NOT gate
The square root of NOT gate acts on a single qubit. It maps the basis state
$$\displaystyle
to
$$\displaystyle \frac (1+i)2
and
$$\displaystyle
to
$$\displaystyle \frac (1-i)2
.
.ˑ.
$$\displaystyle \sqrt NOT\,\sqrt NOT=NOT
, so this gate is a square root of the NOT gate.
Similar squared root-gates can be constructed for all other gates by finding the unitary matrix that, multiplied by itself, yields the gate one wishes to construct the squared root gate of. All rational exponents of all gates can be created this way. (Exact synthesis may result in a very deep or even infinite gate depth, especially if the exponent is irrational, and so one may wish to settle for an approximation.)
Phase shift (
This is a family of single-qubit gates that leave the basis state
$$0\rangle
unchanged and map
$$1\rangle
to
$$1\rangle
. The probability of measuring a
$$\displaystyle
or
$$1\rangle
is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude) on the Bloch sphere by
$$\displaystyle \phi
radians.
where
$$\displaystyle \phi
is the phase shift. Some common examples are the
$$\displaystyle \frac \pi 8
gate (commonly written as T) where
$$\displaystyle \phi =\frac \pi 4
, the phase gate (written S, though S is sometimes used for SWAP gates) where
$$\displaystyle \phi =\frac \pi 2
and the Pauli-Z gate where
$$\displaystyle \phi =\pi
.
Swap (SWAP) gate[ edit ]
Circuit representation of SWAP gate
The swap gate swaps two qubits. With respect to the basis
$$00\rangle
,
$$01\rangle
,
$$\displaystyle
,
$$11\rangle
, it is represented by the matrix:
Square root of Swap gate (√SWAP)[ edit ]
Circuit representation of
$$\displaystyle \sqrt \mboxSWAP
gate
The sqrt(swap) gate performs half-way of a two-qubit swap. It is universal such that any many-qubit gate can be constructed from only sqrt(swap) and single qubit gates. The sqrt(swap) gate is not, however maximally entangling; more than one application of it is required to produce a Bell state from product states.
Controlled (cX cY cZ) gates[ edit ]
Circuit representation of controlled NOT gate
Hadamard-CNOT combination
Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation. For example, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is
$$\displaystyle
, and otherwise leaves it unchanged. With respect to the basis
$$00\rangle
,
$$01\rangle
,
$$10\rangle
,
$$11\rangle
, it is represented by the matrix:
More generally if U is a gate that operates on single qubits with matrix representation
then the controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.
Circuit representation of controlled-U gate
The matrix representing the controlled U is
When U is one of the Pauli matrices , σ_{x}, σ_{y}, or σ_{z}, the respective terms “controlled-X“, “controlled-Y“, or “controlled-Z” are sometimes used.^{ [1] }
The CNOT gate is generally used in quantum computing to generate entangled states .
Toffoli (CCNOT) gate[ edit ]
Circuit representation of Toffoli gate
The Toffoli gate, named after Tommaso Toffoli ; also called CCNOT gate or Deutsch
$$\displaystyle D(\pi /2)
gate; is a 3-bit gate, which is universal for classical computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If we limit ourselves to only accepting input qubits that are
$$\displaystyle
and
$$1\rangle
, then if the first two bits are in the state
$$1\rangle
it applies a Pauli-X (or NOT) on the third bit, else it does nothing. It is an example of a controlled gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.^{ [2] }
Truth table | Matrix form | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $$ \displaystyle \beginbmatrix1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\endbmatrix |
It can be also described as the gate which maps
$$a,b,c\rangle
to
$$\displaystyle
.
Fredkin (CSWAP) gate[ edit ]
Circuit representation of Fredkin gate
The Fredkin gate (also CSWAP or cS gate), named after Edward Fredkin , is a 3-bit gate that performs a controlled swap . It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.
Truth table | Matrix form | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $$ \displaystyle \beginbmatrix1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\\endbmatrix |
Ising (XX) gate[ edit ]
The Ising gate (or XX gate) is a 2-qubit gate that is implemented natively in some trapped-ion quantum computers.^{ [3] }^{ [4] } It is defined as
Deutsch (
The Deutsch (or
$$\displaystyle D_\theta
) gate, named after physicist David Deutsch is a three-qubit gate. It is defined as
Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. However, a method was proposed to realize such a Deutsch gate with dipole-dipole interaction in neutral atoms.
Universal quantum gates[ edit ]
Both CNOT and
$$\displaystyle \sqrt \mboxSWAP
are universal two-qubit gates and can be transformed into each other.
Informally, a set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible since the number of possible quantum gates is uncountable , whereas the number of finite sequences from a finite set is countable . To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently.
One simple set of two-qubit universal quantum gates is the Hadamard gate
$$\displaystyle H
, the
$$\displaystyle \pi /8
gate
$$\displaystyle R_Z(\pi /4)=diag(1,e^\frac i\pi 4)
, and the controlled-NOT gate
$$\displaystyle cX
.^{ [5] }
A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate
$$\displaystyle D(\theta )
, which performs the transformation^{ [6] }
The universal classical logic gate, the Toffoli gate , is reducible to the Deutsch gate ,
$$\displaystyle D(\beginmatrix\frac \pi 2\endmatrix)
, thus showing that all classical logic operations can be performed on a universal quantum computer.
Another set of universal quantum gates consists of the Ising gate and the phase-shift gate. These are the set of gates natively available in some trapped-ion quantum computers.^{ [4] }
Measurement[ edit ]
Circuit representation of measurement. The two lines on the right hand side represents a classical bit, the single line on the left hand side represents a qubit.
Measurement is irreversible and therefore not a quantum gate, because it assigns the observed variable to a singular value. Measurement takes a quantum state and projects it to one of the base vectors, with a likelihood equal to the square of the vectors depth along that base vector. This is a non-reversible operation as it sets the quantum state equal to the base vector that represents the measured state (the state “collapses” to a definite singular value). Why and how this is so is called the measurement problem .
If two different quantum registers are entangled (they cannot be expressed as a tensor product ), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. An example of such a linearly inseparable state is the EPR pair , which can be constructed with the CNOT and the Hadamard gates (described above). This effect is used in many algorithms: if two variables A and B are maximally entangled (the bell state is the simplest example of this), a function F is applied to A such that A is updated to the value of F(A), followed by measurement of A, then B will, when measured, be a value such that F(B) = A^{[ citation needed ]}. This way, measurement of one register can be used to assign properties to some other registers^{[ citation needed ]}.
This effect of assignment is used in Shor’s algorithm . The algorithm uses two measurements on two registers with entangled copies of a single value that is in a superposition; the first measurement is used to obtain an modular exponentiation and to eliminate all values that do not correspond to this modular exponentiation in the other register. This other register is then fed through a quantum fourier transform and then measured to reveal the period, which concludes the algorithm. The order in which measurement is performed can be reversed, or concurrently interleaved, without affecting the result, since the measurements assignment of one register will limit the value-space from the other entangled register.
This type of value-assignment in theory occurs instantaneously over any distance and this has as of 2018 been experimentally verified for distances of up to 1200 kilometers^{ [7] }^{ [8] }. That the phenomena appears to violate the speed of light is called the EPR paradox and it is an open question in physics how to resolve this. Originally it was solved by giving up the assumption of local realism , but other interpretations have also emerged. For more information see the Bell test experiments .
Circuit composition and entangled states[ edit ]
If two or more qubits are viewed as a single quantum state, this combined state is equal to the tensor product of the constituent qubits. An entangled state is any state that cannot be tensor-factorized (the state cannot be separated into its constituent qubits). The CNOT, Ising and Toffoli gates are examples of gates that act on states constructed of multiple qubits.
The tensor product of two
$$\displaystyle n
-qubit quantum gates generates the gate that is equal to the two gates in parallel. This gate will act on
$$\displaystyle 2n
qubits. For example, the gate
$$\displaystyle G=H\otimes H
is the Hadamard gate (
$$\displaystyle H
) applied in parallel on 2 qubits. It can be written as
$$
\displaystyle G=H\otimes H=\frac 1\sqrt 2\beginbmatrix1&1\\1&-1\endbmatrix\otimes \frac 1\sqrt 2\beginbmatrix1&1\\1&-1\endbmatrix=\frac 12\beginbmatrix1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\endbmatrix
This “two-qubit parallel Hadamard gate” will when applied to, for example, the two-qubit zero-vector (
$$00\rangle
), create a quantum state that have equal probability of being observed in any of its four possible outcomes; 00, 01, 10 and 11. We can write this operation as:
$$
11\rangle =\frac 10\rangle +2
Here the amplitude is
$$\displaystyle \frac 12
and the probability is equal to the amplitude squared. The probability to observe any state is the absolute value of the amplitude squared, which in the above example means that there is one in four that we observe any one of the individual four cases. (Strictly speaking, the probability is equal to the amplitude modulus squared, and therefore must be real and non-negative. For amplitude
$$\displaystyle \psi
, the probability is its modulus squared
$$\psi
.)
If we have a set of N qubits that are entangled (their combined state can not be tensor-factorized into an expression of the individual qubits) and wish to apply a quantum gate on M < N qubits in the set, we will have to extend the gate to take N qubits. This can be done by combining the gate with an identity matrix such that their tensor product becomes a gate that act on N qubits. The identity matrix (
$$\displaystyle I
) is a representation of the gate that maps every state to itself (i.e., does nothing at all). In a circuit diagram the identity gate or matrix will appear as just a wire.
For example, the Hadamard transform (
$$\displaystyle H
) acts on a single qubit, but if we for example feed it the first of the two qubits that constitute the entangled Bell state
$$\displaystyle \frac 00\rangle +\sqrt 2
, we cannot write that operation easily. We need to extend the Hadamard transform with the do-nothing gate
$$\displaystyle I
so that we can act on quantum states that span two qubits:
$$
\displaystyle M=H\otimes I=\frac 1\sqrt 2\beginbmatrix1&1\\1&-1\endbmatrix\otimes \beginbmatrix1&0\\0&1\endbmatrix=\frac 1\sqrt 2\beginbmatrix1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\endbmatrix
The gate
$$\displaystyle M
can now be applied to any two-qubit state, entangled or otherwise. The M-gate will leave the second qubit untouched and apply the Hadamard transform to the first qubit. If applied to the Bell state in our example, we may write that as:
$$
\displaystyle M\frac 11\rangle \sqrt 2=\frac 12\beginbmatrix1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\endbmatrix\beginbmatrix1\\0\\0\\1\endbmatrix=\frac 12\beginbmatrix1\\1\\1\\-1\endbmatrix=\frac 10\rangle -2
Because the number of elements in the matrices is
$$\displaystyle 2^2n
, where
$$\displaystyle n
is the number of qubits the gates act on, it is believed to be intractable to simulate large quantum systems using classical computers.
History[ edit ]
The current notation for quantum gates was developed by Barenco et al.,^{ [9] } building on notation introduced by Feynman.^{ [10] }
See also[ edit ]
- Pauli matrices
- Quantum circuit
Notes[ edit ]
- ^ M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000
- ^ Aharonov, Dorit (2003-01-09). “A Simple Proof that Toffoli and Hadamard are Quantum Universal”. arXiv : quant-ph/0301040 .
- ^ http://online.kitp.ucsb.edu/online/mbl_c15/monroe/pdf/Monroe_MBL15Conf_KITP.pdf
- ^ ^{a} ^{b} http://iontrap.umd.edu/wp-content/uploads/2012/12/nature18648.pdf
- ^ M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2016, p. 189; ISBN 978-1-107-00217-3
- ^ Deutsch, David (September 8, 1989), “Quantum computational networks”, Proc. R. Soc. Lond. A, 425 (1989): 73–90, Bibcode : 1989RSPSA.425…73D , doi : 10.1098/rspa.1989.0099
- ^ https://www.scientificamerican.com/article/china-shatters-ldquo-spooky-action-at-a-distance-rdquo-record-preps-for-quantum-internet/
- ^ https://www.sciencemag.org/news/2017/06/china-s-quantum-satellite-achieves-spooky-action-record-distance
- ^ Phys. Rev. A 52 3457–3467 (1995), doi : 10.1103/PhysRevA.52.3457 ; e-print arXiv : quant-ph/9503016
- ^ R. P. Feynman, “Quantum mechanical computers”, Optics News, February 1985, 11, p. 11; reprinted in Foundations of Physics 16(6) 507–531.
References[ edit ]
- M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000
- Quantum gates
- Quantum information science
- Logic gates
- Articles needing additional references from February 2018
- All articles needing additional references
- Wikipedia articles needing clarification from February 2018
- All articles with unsourced statements
- Articles with unsourced statements from February 2018
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Why Do Multiple Hadamard Gates Return to Base State?
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Reading through IBM’s intro to quantum computing and gates and I’m confused about the Hadamard gate. When you use an H gate it appears to rotate the qubit along the X axis pi/4 putting it in a superposition.
If my base state was |0> and I pass it through an H gate it will be put into |>. Another H gate will put it back into |0>.
If my base state was |1> then the same happens, where two H gates will return to |1>.
I’d expect two H gates would be the same as a NOT gate and flip |0> to |1>.
I see what is happening but I don’t know why. Why doesn’t two H gates turn |0> to |1> ?
- What do you mean by "|>"?
– Nathaniel
Oct 7 ’17 at 6:04 - 2If by rotations, you refer to the Bloch sphere representation, the Hadamard gate is a rotation of $\pi$ about the vector $\hatx+\hatz$, not $\pi/4$ about the vector $\hatx$ as you stated.
– user154997
Oct 7 ’17 at 6:39
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In a nutshell, there are multiple ways of being in a superposition of |0> and |1>.
Take a look at the explanation of the Hadamard gate found here:
https://developer.ibm.com/dwblog/2016/quantum-computing-everyone-programmers-perspective/
In the graphic representation, all states “reflect” across the same dotted line. When reflected twice, they go back where they came from. You can see that |0> and |1> are reflected to different places, but that both places have equal probability of measuring in each state.
The most clear mathematical representation is to look at the Hadamard gate as a 2×2 matrix 1/$\sqrt2$ [[1,1],[1,-1]]. Square it and you get unity. It might also help to look into the Bloch sphere:
Understanding the Hadamard gate on the Bloch sphere
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Linked
Visual interpretation, on the Bloch sphere, when Hadamard gate is applied twice
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Intuition behind the Hadamard gate
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I’m trying to teach myself about quantum computing, and I have a decent-ish understanding of linear algebra.
I got through the NOT gate, which wasn’t too bad, but then I got to the Hadamard gate. And I got stuck. Mainly because while I “understand” the manipulations, I don’t understand what they really do or why you’d want to do them, if that makes sense.
For example, when the Hadamard gate takes in $|0\rangle$ it gives $\frac\sqrt2$. What does this mean? For the NOT gate, it takes in $|0\rangle$ and gives $|1\rangle$. Nothing unclear about that; it gives the “opposite” of the bit (for the superposition, it takes in $\alpha|0\rangle+\beta|1\rangle$ and gives $\beta|0\rangle + \alpha|1\rangle$) and I understand why that is useful; for the same reasons (basically) that it is useful in a classical computer. But what (for example) is the Hadamard gate doing geometrically to a vector $\beginbmatrix\alpha \\ \beta \endbmatrix$? And why is this useful?
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The Hadamard gate might be your first encounter with superposition creation. When you say you can relate the usefulness of the Pauli $X$ gate (a.k.a. NOT
) to its classical counterpart – well, Hadamard is exactly where you leave the realm of classical analogue, then. It is useful for exactly the same reason, however, namely that it is often used to form a universal set of gates (like clasical AND
with NOT
and fan-out, or NOR
with fan-out alone).
While a single $H$ gate is somewhat directly useful in random number generation (as Yuval Filmus said), its true power shows when appearing in more instances or in combination with other gates. When you have $n$ qubits initialized in $|0\rangle$, for example, and apply one $H$ to each of them in any order, what you get is
$$(|0\rangle + |1\rangle) \otimes (|0\rangle + |1\rangle) \otimes \ldots \otimes (|0\rangle + |1\rangle) / 2^n/2$$
which can be expanded to
$$1/2^n/2 \cdot (|00\ldots00\rangle + |00\ldots01\rangle + |00\ldots11\rangle + \ldots + |11\ldots11\rangle)$$
Voilà, we can now evaluate functions on $2^n$ different inputs in parallel! This is, for example, the first step in Grover’s algorithm .
Another popular use is a Hadamard on one qubit followed by a CNOT
controlled with the qubit you just put into a superposition. See:
$$CNOT \big(2^-1/2(|0\rangle+|1\rangle)\otimes|0\rangle \big) = 2^-1/2 CNOT(|00\rangle + |10\rangle) = 2^-1/2 (|00\rangle + |11\rangle)$$
That’s a Bell state which is a cornerstone of various quantum key distribution protocols, measurement-based computation , quantum teleportation and many more applications. You can also use a CNOT
repeatedly on more zero-initialized target qubits (with the same control) to create
$$2^-1/2 (|00\ldots00\rangle + |11\ldots11\rangle)$$
which is known as the GHZ state , also immensely useful.
Last but not least, it’s a quite useful basis transform that is self-reversible. So another Hadamard gate undoes, in a sense, what a previous application did ($H^2 = I$). You can experiment around what happens if you use it to “sandwich” other operations, for example put one on the target qubit of a CNOT
gate and another after it. Or on both of the qubits (for a total of 4 Hadamards). Try it yourself and you’ll certainly learn a lot about Quantum computation!
Re “what is the Hadamard gate doing geometrically to a vector”: read up on the Bloch sphere , you’ll going to hear about it everywhere. In this representation, a Hadamard gate does a 180° rotation about a certain slanted axis. The Pauli gates (NOT
being one out of three) also do 180° rotations but only about $x$ or $y$ or $z$. Because such geometrical operations are quite restricted, these gates alone can’t really do much. (Indeed, if you restrict yourself to those and a CNOT
in your quantum computer, you just build a very expensive and uneffective classical device.) Rotating about something tilted is important, and one more ingredient you usually need is also rotating by a smaller fraction of the angle, like 45° (like in the Phase shift gate ).
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The Hadamard gate operates on a single qubit. The state of a single qubit can be described as $\alpha \left|0\right\rangle + \beta \left|1\right\rangle$, where $|\alpha|^2 + |\beta|^2 = 1$. If you measure the qubit, the output is $0$ with probability $|\alpha|^2$, and $1$ with probability $|\beta|^2$. From a linear-algebraic perspective, the state of a qubit is just a unit norm vector of length two over the complex numbers. The two vectors $\left|0\right\rangle,\left|1\right\rangle$ span a vector space of dimension two (over the complex numbers), and every unit norm vector in that vector space can be the state of a qubit.
Since the state always has unit norm, the only linear operators possible on qubits are those that preserve norms. From linear algebra, we know that these are exactly the Hermitian operators. To describe an operator, it suffice to describe its effect on a basis. For example, the value of your operator on the vector $\left| 0 \right\rangle$ is $\frac\left\sqrt2$.
According to Wikipedia , the Hadamard gate is used to form a “random input”. If applied to a constant qubit (i.e., $\left| 0 \right\rangle$, $\left| 1 \right\rangle$, or a rotation of these by a unit norm complex number), the Hadamard gate forms a “uniformly random” qubit, which when measured behaves like a fair coin toss. This is the kind of behavior we want when “trying all possibilities in parallel”.
I suggest you continue your reading on quantum computation; when you get to quantum algorithms (like Grover’s and Shor’s), you will understand what all these quantum gates are useful for.
- 2"unit norm vector of length two" was confusing to me because I’m used to using norm and length interchangeably.
– adrianN
Sep 26 ’16 at 8:36
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