This is the kind of application that often requires a buck/boost transformer setup.

Using a 240 V feed and a transformer at the remote gate to drive 120 V equipment works well.

At 240 V the current is half, cutting the voltage drop in half.

To elaborate further on the post by brickeyee, for a given wattage load on a given conductor, the voltage drop at 240 volts will be 1/4 the drop that will be experienced if the system is at 120 volts. As a practical matter, if an induction motor will start and power the load, the tolerable voltage drop can be greater if the load is of short duration with significant cooling periods between loads. That seems to fit the gate application.

"for a given wattage load on a given conductor, the voltage drop at 240 volts will be 1/4 the drop that will be experienced if the system is at 120 volts."

Don't think so.

At the same wattage (say 240 watts) 240 V is half the current (1 amp vs. 2 amps at 120 V) so half the voltage drop.
The presence of a neutral does not affect the drop in a 240 V circuit since the neutral is not used.

The POWER lost will be 1/4.

Either way, wiring for 240v instead of 120v will let you reduce the size of the wire and save some money. You can then choose to use a transformer to give you 120v at the gate, or perhaps look for 240v lamps and gate equipment and wire them directly.

I don't know exactly what style of gate & opener the OP is considering, but I think the power requirements are being greatly over-estimated.

I have a 16' swing gate at the end of our driveway with a remote controlled opener. Power requirements on the opener's label are 12 VDC @ 25 watts when operating - which is supplied by the internal rechargeable sealed lead acid battery rated 12V 7.0Ah. For most of the year it's only charge is received from it's 10 watt solar panel. This time of year (3-4 months) I supplement it with it's 18VAC/40VA class 2 transformer for a few hours (on timer) each day.

The driveway is about 600' but since it curves as it comes up, a straight feed from the house across the property reduced an underground run to around 400'. I ran 12 gauge in conduit (gophers!) to power two 13 watt CFL lamp post fixtures on photocells and an outlet for the gate transformer. With both the lights and transformer on it measures 34 watts.

The opener is a Mighty Mule similar to the current model 500 and can handle gates up to 850 lbs depending on length. I'm including a link to the manufactures website as a reference.

Here is a link that might be useful: Mighty Mule gate openers

A lot depends on the size and weight of the gates, and even hinge quality (hinges with actual bearings can greatly reduce the opening force required).

I had a similar situation - 600ft drivway and wanted a gate. I wasn't sure what I was going to need for the gate when the house was being built so when the well trench was dug (which is about 100 ft from the drive entrance), I had them drop a #10 AWG direct burial cable into the trench and bring it into the house. That allowed me a 6 amp load at the gate with an acceptable voltage drop - more than enough to power a gate. I also dropped in a direct burial phone cable, TV cable, and a 4 conductor 16 gauge low voltage cable. Turns out for me that all of these were kind of important to have.

I installed a lightweight gate made my Amazing gates. It is a double gate with a 16 ft span and each leaf weighs around 130 lbs. I went this route because I could install it myslef easily and it really looks like an iron gate.

The power for the gate actuators are driven off of a battery in the gate controller housing and is low voltage DC. The battery trickle charges and that can be done with either a direct 24V AC power from the house or from a 120V AC source at the gate which is what I used. Heavier gates require more power and a 120V may be the only way to make those work - you need to make that decision before your drop your cable.

What i did not realize - and luckily had the cable to do it, was that I also wanted a call box so that people could buzz me when they wanted to come in OR the gas company could let themselves in when I am not home if I provide them an access code (and this is a big deal for me - I am never home when gas company shows). Power for the call box comes from the 120V outlet powering the gate controller but the box itself requires 24V DC. The way the call box works is that you must run a phone line from the house to the box and then back (you need 4 conductors). That allows the call box to seize the phone line and create a special ring tone on my house phone so I know someone is at the gate. I can then answer the phone and buzz them in. It also allows the gate box to call me on my cell phone (if there is no answer after 4 rings) and I can talk to the person at the gate and buzz them in remotely - very cool and handy! Just make sure you set your house answering machine to pick up after the gate box (e.g. gate box set to 4 rings, house answering machine set to 6 rings)

The call box also has an optional video camera which i have not installed yet but might do in the future. If you want to see who is at the gate, you will need a shielded video cable and seperate LV power to drive the camera.

Someday in the near future I also plan to add an lighted address sign near the gate so the 120V feed gives me the flexibility to power that.

Finally, I added the 120V gate circuit onto my power generator backup so that when the power goes out, my gate battery still charges.

You can do it all with low voltage wire but I think having the 120V outlet at the front provides more flexibility and easy install of the gate controller. Of course there is the cost factor - the cable wasn't cheap.

There are online calculators that willl tell you the maximum voltage drop for a given cable length, guage and load. You can balance your power needs vs cable size and cost. Then just make sure you don't plug in a device that exceeds the max load.

I suggest picking out your gate controller and call box now and downloading the installation instructions (you don't have to buy it - just get the documents). It will have all of the wiring requirements for the gate. There are quite a few add ons that you will need - like Firebox (so the fire dept can open the gate if you aren't home and the house is ablaze), automatic gate opener for exiting vehicles , and IR beams for safety to make sure that the gate won't close on a car that stops in the closure path. There is a lot more wiring a gate than the power.

Good luck with the project.

Extra wires and foresight are a great thing. I was also going to suggest considering using aluminium wire with the appropriate fittings/paste etc, you can then use heavier gauge for less money. Round here, all subpanels et al are interconnected with armoured aluminium.